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<title>Exponentiation</title>

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Exponentiation

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</style><div role="note" class="hatnote navigation-not-searchable">"Exponent" redirects here. For other uses, see <a href="Exponent_(disambiguation)" class="mw-disambig" title="Exponent (disambiguation)">Exponent (disambiguation)</a>.</div>

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</style><table class="sidebar nomobile nowraplinks"><tbody><tr><td class="sidebar-above" style="background:var(--background-color-neutral, #efefef);">

<a href="Arithmetic_operations" class="mw-redirect" title="Arithmetic operations">Arithmetic operations</a></td></tr><tr><td class="sidebar-content" style="font-size:130%;">

<table class="infobox-subbox infobox-3cols-child infobox-table"><tbody><tr><th colspan="4" class="infobox-header"><a href="Addition" title="Addition">Addition</a> (+)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;">

<mrow>

<mtr>

<mtd>

</mrow>

</mrow>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mtext>summand</mtext>

</mrow>

<mtext>summand</mtext>

</mrow>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mtext>addend</mtext>

</mrow>

<mtext>addend</mtext>

</mrow>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mtext>augend</mtext>

</mrow>

<mtext>addend</mtext>

</mrow>

</mstyle>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}</annotation>

</semantics>

</math><img src="./ea99a27b5a763ef48889c450ac8157083ea97118.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:20.217ex; height:9.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}" loading="lazy"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{sum}}}</annotation>

</semantics>

</math><img src="./b8609baca9fdbc4c529f5894884a08122d695dad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.931ex; height:1.343ex;" alt="{\displaystyle \scriptstyle {\text{sum}}}" loading="lazy"></td></tr><tr><th colspan="4" class="infobox-header"><a href="Subtraction" title="Subtraction">Subtraction</a> (−)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;">

<mrow>

<mtr>

<mtd>

</mrow>

</mrow>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mtext>minuend</mtext>

</mrow>

<mtext>subtrahend</mtext>

</mrow>

</mstyle>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}</annotation>

</semantics>

</math><img src="./2780b756445a5f8f95b16c33e3b924f976958ea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.356ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}" loading="lazy"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;">

<mtext>difference</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{difference}}}</annotation>

</semantics>

</math><img src="./1ac22c4e24eef2036cff5bfea924cc0dbb30c5d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.857ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {\text{difference}}}" loading="lazy"></td></tr><tr><th colspan="4" class="infobox-header"><a href="Multiplication" title="Multiplication">Multiplication</a> (×)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;">

<mrow>

<mtr>

<mtd>

<mtext>factor</mtext>

</mrow>

<mtext>factor</mtext>

</mrow>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mtext>multiplier</mtext>

</mrow>

<mtext>multiplicand</mtext>

</mrow>

</mstyle>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}</annotation>

</semantics>

</math><img src="./93f7b476e32221c7b05d356289c8085aef54059b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.176ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}" loading="lazy"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;">

<mtext>product</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{product}}}</annotation>

</semantics>

</math><img src="./a5c8b7509b8be1043622cb7b1b9a36ca8bfc2616.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.578ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\text{product}}}" loading="lazy"></a></td></tr><tr><th colspan="4" class="infobox-header"><a href="Division_(mathematics)" title="Division (mathematics)">Division</a> (÷)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;">

<mrow>

<mtr>

<mtd>

<mfrac>

<mtext>dividend</mtext>

</mrow>

</mstyle>

<mtext>divisor</mtext>

</mrow>

</mstyle>

</mfrac>

</mrow>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mtext>numerator</mtext>

</mrow>

</mstyle>

<mtext>denominator</mtext>

</mrow>

</mstyle>

</mfrac>

</mrow>

</mstyle>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}</annotation>

</semantics>

</math><img src="./5d5d22ff59234f0d437be740306e8dd905991e1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.15ex; height:8.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}" loading="lazy"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;">

<mrow>

<mtr>

<mtd>

<mtext>fraction</mtext>

</mrow>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mtext>quotient</mtext>

</mrow>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<mtext>ratio</mtext>

</mrow>

</mstyle>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}</annotation>

</semantics>

</math><img src="./2359c3ca6e50e7ae8065baa710440b3c79895023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:8.197ex; height:7.176ex;" alt="{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}" loading="lazy"></a></td></tr><tr><th colspan="4" class="infobox-header"></th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;">

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

<mtext>exponent</mtext>

</mrow>

</msup>

</mstyle>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

<mtext>power</mtext>

</mrow>

</msup>

</mstyle>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}</annotation>

</semantics>

</math><img src="./ecb107371002b62a60fcbd13e742f4d81f872b67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.618ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}" loading="lazy"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;">

<mtext>power</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{power}}}</annotation>

</semantics>

</math><img src="./b0d0a9fbffb659c0055d5ee6fde3f7f28e96f45c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.297ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {\text{power}}}" loading="lazy"></td></tr><tr><th colspan="4" class="infobox-header"><a href="Nth_root" title="Nth root">nth root</a> (√)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;">

<mroot>

<mtext>radicand</mtext>

</mrow>

</mstyle>

<mtext>degree</mtext>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}</annotation>

</semantics>

</math><img src="./5582d567e7e7fbcdb728291770905e09beb0ea18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.422ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}" loading="lazy"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{root}}}</annotation>

</semantics>

</math><img src="./2a015c1122190da3f1f1732d88b8bb03a8d7eb91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.928ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {\text{root}}}" loading="lazy"></td></tr><tr><th colspan="4" class="infobox-header"><a href="Logarithm" title="Logarithm">Logarithm</a> (log)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;">

<msub>

</mrow>

</msub>

<mtext>anti-logarithm</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}</annotation>

</semantics>

</math><img src="./2435266fcae4aa91d3d70a74bb91b5b35ef52edd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.454ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}" loading="lazy"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;">

<mtext>logarithm</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{logarithm}}}</annotation>

</semantics>

</math><img src="./fe5d50baa86b950ff6d15760b7a38df1f8d8c868.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.948ex; height:2.009ex;" alt="{\displaystyle \scriptstyle {\text{logarithm}}}" loading="lazy"></td></tr></tbody></table></td>

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In <a href="Mathematics" title="Mathematics">mathematics</a>, exponentiation, denoted bn, is an <a href="Operation_(mathematics)" title="Operation (mathematics)">operation</a> involving two numbers: the base, b, and the exponent or power, n.<a href="#cite_note-:1-1">[1]</a> When n is a positive <a href="Integer" title="Integer">integer</a>, exponentiation corresponds to repeated <a href="Multiplication" title="Multiplication">multiplication</a> of the base: that is, bn is the <a href="Product_(mathematics)" title="Product (mathematics)">product</a> of multiplying n bases:<a href="#cite_note-:1-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle b^{n}=\underbrace {b\times b\times \dots \times b\times b} _{n{\text{ times}}}.}</annotation>

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</math>In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{1}=b}">

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<annotation encoding="application/x-tex">{\displaystyle b^{1}=b}</annotation>

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</math><img src="./7d240dbaf6181ae1801474f3d28dcd5504aacae6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.148ex; height:2.676ex;" alt="{\displaystyle b^{1}=b}" loading="lazy">.

The exponent is usually shown as a <a href="Superscript" class="mw-redirect" title="Superscript">superscript</a> to the right of the base as bn or in computer code as <code>b^n</code>. This <a href="Binary_operation" title="Binary operation">binary operation</a> is often read as "b to the power n"; it may also be referred to as "b raised to the nth power", "the nth power of b",<a href="#cite_note-2">[2]</a> or, most briefly, "b to the n".

The above definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle b^{n}}</annotation>

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</math><img src="./b8f8f52cd26bb201e02c8d1b3619a3a682f44dbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.216ex; height:2.343ex;" alt="{\displaystyle b^{n}}" loading="lazy"> immediately implies several properties, in particular the multiplication rule:<a href="#cite_note-3">[nb 1]</a>

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<mtext> times</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}b^{n}\times b^{m}&=\underbrace {b\times \dots \times b} _{n{\text{ times}}}\times \underbrace {b\times \dots \times b} _{m{\text{ times}}}\\[1ex]&=\underbrace {b\times \dots \times b} _{n+m{\text{ times}}}\ =\ b^{n+m}.\end{aligned}}}</annotation>

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That is, when multiplying a base raised to one power times the same base raised to another power, the powers add. Extending this rule to the power zero gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{0}\times b^{n}=b^{0+n}=b^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle b^{0}\times b^{n}=b^{0+n}=b^{n}}</annotation>

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</math><img src="./530a1a0a7fbb93881bc111cd2b85bb10dcdec341.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.838ex; height:2.676ex;" alt="{\displaystyle b^{0}\times b^{n}=b^{0+n}=b^{n}}" loading="lazy">, and, where b is non-zero, dividing both sides by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle b^{n}}</annotation>

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</math><img src="./b8f8f52cd26bb201e02c8d1b3619a3a682f44dbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.216ex; height:2.343ex;" alt="{\displaystyle b^{n}}" loading="lazy"> gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{0}=b^{n}/b^{n}=1}">

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<annotation encoding="application/x-tex">{\displaystyle b^{0}=b^{n}/b^{n}=1}</annotation>

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</math><img src="./ba6e097290fe5f92dc7836ab3dac8d984d91a10b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.006ex; height:3.176ex;" alt="{\displaystyle b^{0}=b^{n}/b^{n}=1}" loading="lazy">. That is the multiplication rule implies the definition <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{0}=1.}">

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<annotation encoding="application/x-tex">{\displaystyle b^{0}=1.}</annotation>

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</math>A similar argument implies the definition for negative integer powers: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{-n}=1/b^{n}.}">

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<annotation encoding="application/x-tex">{\displaystyle b^{-n}=1/b^{n}.}</annotation>

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</math>That is, extending the multiplication rule gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{-n}\times b^{n}=b^{-n+n}=b^{0}=1}">

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<annotation encoding="application/x-tex">{\displaystyle b^{-n}\times b^{n}=b^{-n+n}=b^{0}=1}</annotation>

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</math><img src="./fe9db3eb6fb14fbc7a4ac0df4ab887eee63b344d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:26.82ex; height:2.676ex;" alt="{\displaystyle b^{-n}\times b^{n}=b^{-n+n}=b^{0}=1}" loading="lazy">. Dividing both sides by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle b^{n}}</annotation>

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</math><img src="./b8f8f52cd26bb201e02c8d1b3619a3a682f44dbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.216ex; height:2.343ex;" alt="{\displaystyle b^{n}}" loading="lazy"> gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{-n}=1/b^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle b^{-n}=1/b^{n}}</annotation>

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</math><img src="./2bc089b4a3b2678e49e8a5edef0b8ac3d148f631.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.134ex; height:3.009ex;" alt="{\displaystyle b^{-n}=1/b^{n}}" loading="lazy">. This also implies the definition for fractional powers: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{n/m}={\sqrt[{m}]{b^{n}}}.}">

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<annotation encoding="application/x-tex">{\displaystyle b^{n/m}={\sqrt[{m}]{b^{n}}}.}</annotation>

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</math>For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{1/2}\times b^{1/2}=b^{1/2\,+\,1/2}=b^{1}=b}">

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<annotation encoding="application/x-tex">{\displaystyle b^{1/2}\times b^{1/2}=b^{1/2\,+\,1/2}=b^{1}=b}</annotation>

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</math><img src="./67d2aa8ea99a21dea3315b33c4d600815df9a208.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:30.791ex; height:2.843ex;" alt="{\displaystyle b^{1/2}\times b^{1/2}=b^{1/2\,+\,1/2}=b^{1}=b}" loading="lazy">, meaning <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b^{1/2})^{2}=b}">

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<annotation encoding="application/x-tex">{\displaystyle (b^{1/2})^{2}=b}</annotation>

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</math><img src="./9bf168d956d455319cc4259148a8253d66a4425d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.655ex; height:3.343ex;" alt="{\displaystyle (b^{1/2})^{2}=b}" loading="lazy">, which is the definition of square root: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{1/2}={\sqrt {b}}}">

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<annotation encoding="application/x-tex">{\displaystyle b^{1/2}={\sqrt {b}}}</annotation>

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</math><img src="./949cc58a0bd98ac174ea45a99c0905771cd0bff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.728ex; height:3.176ex;" alt="{\displaystyle b^{1/2}={\sqrt {b}}}" loading="lazy">.

The definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{x}}">

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<annotation encoding="application/x-tex">{\displaystyle b^{x}}</annotation>

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</math><img src="./2bb7406a338fb530330582bc63420d091897c709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.17ex; height:2.343ex;" alt="{\displaystyle b^{x}}" loading="lazy"> for any positive real base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> and any real number exponent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. More involved definitions allow <a href="Complex_numbers" class="mw-redirect" title="Complex numbers">complex</a> base and exponent, as well as certain types of <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> as base or exponent.

Exponentiation is used extensively in many fields, including <a href="Economics" title="Economics">economics</a>, <a href="Biology" title="Biology">biology</a>, <a href="Chemistry" title="Chemistry">chemistry</a>, <a href="Physics" title="Physics">physics</a>, and <a href="Computer_science" title="Computer science">computer science</a>, with applications such as <a href="Compound_interest" title="Compound interest">compound interest</a>, <a href="Population_growth" title="Population growth">population growth</a>, <a href="Chemical_reaction_kinetics" class="mw-redirect" title="Chemical reaction kinetics">chemical reaction kinetics</a>, <a href="Wave" title="Wave">wave</a> behavior, and <a href="Public-key_cryptography" title="Public-key cryptography">public-key cryptography</a>.

<div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2></div>

The term exponent originates from the <a href="Latin" title="Latin">Latin</a> exponentem, the <a href="Present_participle" class="mw-redirect" title="Present participle">present participle</a> of exponere, meaning "to put forth".<a href="#cite_note-4">[3]</a> The term power (<a href="Latin_language" class="mw-redirect" title="Latin language">Latin</a>: potentia, potestas, dignitas) is a mistranslation<a href="#cite_note-Rotman-5">[4]</a><a href="#cite_note-6">[5]</a> of the <a href="Ancient_Greek" title="Ancient Greek">ancient Greek</a> δύναμις (dúnamis, here: "amplification"<a href="#cite_note-Rotman-5">[4]</a>) used by the <a href="Greek_mathematics" class="mw-redirect" title="Greek mathematics">Greek</a> mathematician <a href="Euclid" title="Euclid">Euclid</a> for the square of a line,<a href="#cite_note-MacTutor-7">[6]</a> following <a href="Hippocrates_of_Chios" title="Hippocrates of Chios">Hippocrates of Chios</a>.<a href="#cite_note-8">[7]</a>

The word exponent was coined in 1544 by Michael Stifel.<a href="#cite_note-9">[8]</a><a href="#cite_note-10">[9]</a> In the 16th century, <a href="Robert_Recorde" title="Robert Recorde">Robert Recorde</a> used the terms "square", "cube", "zenzizenzic" (<a href="Fourth_power" title="Fourth power">fourth power</a>), "sursolid" (<a href="Fifth_power_(algebra)" title="Fifth power (algebra)">fifth</a>), "zenzicube" (<a href="Sixth_power" title="Sixth power">sixth</a>), "second sursolid" (<a href="Seventh_power" title="Seventh power">seventh</a>), and "<a href="Zenzizenzizenzic" title="Zenzizenzizenzic">zenzizenzizenzic</a>" (<a href="Eighth_power" title="Eighth power">eighth</a>).<a href="#cite_note-worldwidewords-11">[10]</a> "Biquadrate" has been used to refer to the fourth power as well.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

In <a href="The_Sand_Reckoner" title="The Sand Reckoner">The Sand Reckoner</a>, <a href="Archimedes" title="Archimedes">Archimedes</a> proved the law of exponents, 10a · 10b = 10a+b, necessary to manipulate powers of 10.<a href="#cite_note-12">[11]</a> He then used powers of 10 to estimate the number of grains of sand that can be contained in the universe.

In the 9th century, the Persian mathematician <a href="Al-Khwarizmi" title="Al-Khwarizmi">Al-Khwarizmi</a> used the terms مَال (māl, "possessions", "property") for a <a href="Square_(algebra)" title="Square (algebra)">square</a>—the Muslims, "like most mathematicians of those and earlier times, thought of a squared number as a depiction of an area, especially of land, hence property"<a href="#cite_note-worldwidewords-11">[10]</a>—and كَعْبَة (<a href="Kaaba" title="Kaaba">Kaʿbah</a>, "cube") for a <a href="Cube_(algebra)" title="Cube (algebra)">cube</a>, which later <a href="Mathematics_in_the_medieval_Islamic_world" title="Mathematics in the medieval Islamic world">Islamic</a> mathematicians represented in <a href="Mathematical_notation" title="Mathematical notation">mathematical notation</a> as the letters <a href="M%C4%ABm" class="mw-redirect" title="Mīm">mīm</a> (m) and <a href="K%C4%81f" class="mw-redirect" title="Kāf">kāf</a> (k), respectively, by the 15th century, as seen in the work of <a href="Abu'l-Hasan_ibn_Ali_al-Qalasadi" title="Abu'l-Hasan ibn Ali al-Qalasadi">Abu'l-Hasan ibn Ali al-Qalasadi</a>.<a href="#cite_note-13">[12]</a>

<a href="Nicolas_Chuquet" title="Nicolas Chuquet">Nicolas Chuquet</a> used a form of exponential notation in the 15th century, for example 122 to represent 12x2.<a href="#cite_note-14">[13]</a> This was later used by <a href="Henricus_Grammateus" title="Henricus Grammateus">Henricus Grammateus</a> and <a href="Michael_Stifel" title="Michael Stifel">Michael Stifel</a> in the 16th century. In the late 16th century, <a href="Jost_B%C3%BCrgi" title="Jost Bürgi">Jost Bürgi</a> would use Roman numerals for exponents in a way similar to that of Chuquet, for example iii4 for 4x3.<a href="#cite_note-cajori-15">[14]</a>

In 1636, <a href="James_Hume_(mathematician)" title="James Hume (mathematician)">James Hume</a> used in essence modern notation, when in L'algèbre de Viète he wrote Aiii for A3.<a href="#cite_note-16">[15]</a> Early in the 17th century, the first form of our modern exponential notation was introduced by <a href="Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> in his text titled <a href="La_G%C3%A9om%C3%A9trie" title="La Géométrie">La Géométrie</a>; there, the notation is introduced in Book I.<a href="#cite_note-17">[16]</a>

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</style><blockquote class="templatequote">I designate ... aa, or a2 in multiplying a by itself; and a3 in multiplying it once more again by a, and thus to infinity.</blockquote><div class="templatequotecite">— René Descartes, <a href="La_G%C3%A9om%C3%A9trie" title="La Géométrie">La Géométrie</a></div>

Some mathematicians (such as Descartes) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write <a href="Polynomial" title="Polynomial">polynomials</a>, for example, as ax + bxx + cx3 + d.

<a href="Samuel_Jeake" title="Samuel Jeake">Samuel Jeake</a> introduced the term indices in 1696.<a href="#cite_note-MacTutor-7">[6]</a> The term involution was used synonymously with the term indices, but had declined in usage<a href="#cite_note-18">[17]</a> and should not be confused with <a href="Involution_(mathematics)" title="Involution (mathematics)">its more common meaning</a>.

In 1748, <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> introduced variable exponents, and, implicitly, non-integer exponents by writing:<blockquote class="templatequote">Consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not <a href="Algebraic_function" title="Algebraic function">algebraic functions</a>, since in those the exponents must be constant.<a href="#cite_note-Euler_1748-19">[18]</a></blockquote>

<div class="mw-heading mw-heading3"><h3 id="20th_century">20th century</h3></div>

As calculation was mechanized, notation was adapted to numerical capacity by conventions in exponential notation. For example <a href="Konrad_Zuse" title="Konrad Zuse">Konrad Zuse</a> introduced <a href="Floating-point_arithmetic" title="Floating-point arithmetic">floating-point arithmetic</a> in his 1938 computer Z1. One <a href="Register_(computer)" class="mw-redirect" title="Register (computer)">register</a> contained representation of leading digits, and a second contained representation of the exponent of 10. Earlier <a href="Leonardo_Torres_Quevedo" title="Leonardo Torres Quevedo">Leonardo Torres Quevedo</a> contributed Essays on Automation (1914) which had suggested the floating-point representation of numbers. The more flexible <a href="Decimal_floating-point" class="mw-redirect" title="Decimal floating-point">decimal floating-point</a> representation was introduced in 1946 with a <a href="Bell_Laboratories" class="mw-redirect" title="Bell Laboratories">Bell Laboratories</a> computer. Eventually educators and engineers adopted <a href="Scientific_notation" title="Scientific notation">scientific notation</a> of numbers, consistent with common reference to <a href="Order_of_magnitude" title="Order of magnitude">order of magnitude</a> in a <a href="Ratio_scale" class="mw-redirect" title="Ratio scale">ratio scale</a>.<a href="#cite_note-20">[19]</a>

For instance, in 1961 the <a href="School_Mathematics_Study_Group" title="School Mathematics Study Group">School Mathematics Study Group</a> developed the notation in connection with units used in the <a href="Metric_system" title="Metric system">metric system</a>.<a href="#cite_note-21">[20]</a><a href="#cite_note-22">[21]</a>

Exponents also came to be used to describe <a href="Units_of_measurement" class="mw-redirect" title="Units of measurement">units of measurement</a> and <a href="Quantity_dimension" class="mw-redirect" title="Quantity dimension">quantity dimensions</a>. For instance, since <a href="Force" title="Force">force</a> is mass times acceleration, it is measured in kg m/sec2. Using M for mass, L for length, and T for time, the expression M L T–2 is used in <a href="Dimensional_analysis" title="Dimensional analysis">dimensional analysis</a> to describe force.<a href="#cite_note-23">[22]</a><a href="#cite_note-24">[23]</a>

<div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2></div>

The expression b2 = b · b is called "the <a href="Square_(algebra)" title="Square (algebra)">square</a> of b" or "b squared", because the area of a square with side-length b is b2. (It is true that it could also be called "b to the second power", but "the square of b" and "b squared" are more traditional)

Similarly, the expression b3 = b · b · b is called "the <a href="Cube_(algebra)" title="Cube (algebra)">cube</a> of b" or "b cubed", because the volume of a cube with side-length b is b3.

When an exponent is a <a href="Positive_integer" class="mw-redirect" title="Positive integer">positive integer</a>, that exponent indicates how many copies of the base are multiplied together. For example, 35 = 3 · 3 · 3 · 3 · 3 = 243. The base 3 appears 5 times in the multiplication, because the exponent is 5. Here, 243 is the 5th power of 3, or 3 raised to the 5th power.

The word "raised" is usually omitted, and sometimes "power" as well, so 35 can be simply read "3 to the 5th", or "3 to the 5".

<div class="mw-heading mw-heading2"><h2 id="Integer_exponents">Integer exponents </h2></div>

The exponentiation operation with integer exponents may be defined directly from elementary <a href="Arithmetic_operation" class="mw-redirect" title="Arithmetic operation">arithmetic operations</a>.

<div class="mw-heading mw-heading3"><h3 id="Positive_exponents">Positive exponents</h3></div>

The definition of the exponentiation as an iterated multiplication can be <a href="Formal_proof" title="Formal proof">formalized</a> by using <a href="Mathematical_induction" title="Mathematical induction">induction</a>,<a href="#cite_note-25">[24]</a> and this definition can be used as soon as one has an <a href="Associativity" class="mw-redirect" title="Associativity">associative</a> multiplication:

The base case is

<msup>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle b^{1}=b}</annotation>

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</math><img src="./7d240dbaf6181ae1801474f3d28dcd5504aacae6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.148ex; height:2.676ex;" alt="{\displaystyle b^{1}=b}" loading="lazy"></dd></dl>

and the <a href="Recurrence_relation" title="Recurrence relation">recurrence</a> is

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle b^{n+1}=b^{n}\cdot b.}</annotation>

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</math><img src="./22becb6fbb370b056af0dc723f2af7e4db6a034a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.955ex; height:2.676ex;" alt="{\displaystyle b^{n+1}=b^{n}\cdot b.}" loading="lazy"></dd></dl>

The associativity of multiplication implies that for any positive integers m and n,

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<annotation encoding="application/x-tex">{\displaystyle b^{m+n}=b^{m}\cdot b^{n},}</annotation>

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</math><img src="./7c0e39ac82e77ae079a3b5ca0d36d94070286a5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.25ex; height:2.843ex;" alt="{\displaystyle b^{m+n}=b^{m}\cdot b^{n},}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle (b^{m})^{n}=b^{mn}.}</annotation>

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</math><img src="./b2814edfd17fe0cbfecbb001b5ce3db1a991b6aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.104ex; height:2.843ex;" alt="{\displaystyle (b^{m})^{n}=b^{mn}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Zero_exponent">Zero exponent</h3></div>

As mentioned earlier, a (nonzero) number raised to the 0 power is 1:<a href="#cite_note-26">[25]</a><a href="#cite_note-:1-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle b^{0}=1.}</annotation>

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</math><img src="./de10f50a06d8e714f55212cadf1da761bcec4c1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.96ex; height:2.676ex;" alt="{\displaystyle b^{0}=1.}" loading="lazy"></dd></dl>

This value is also obtained by the <a href="Empty_product" title="Empty product">empty product</a> convention, which may be used in every <a href="Algebraic_structure" title="Algebraic structure">algebraic structure</a> with a multiplication that has an <a href="Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">identity</a>. This way the formula

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<annotation encoding="application/x-tex">{\displaystyle b^{m+n}=b^{m}\cdot b^{n}}</annotation>

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</math><img src="./259b25c515e091594d2f0ea2b75bf2628ce45f51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.604ex; height:2.509ex;" alt="{\displaystyle b^{m+n}=b^{m}\cdot b^{n}}" loading="lazy"></dd></dl>

also holds for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0}">

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<annotation encoding="application/x-tex">{\displaystyle n=0}</annotation>

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</math><img src="./26819344e55f5e671c76c07c18eb4291fcec85ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=0}" loading="lazy">.

The case of 00 is controversial. In contexts where only integer powers are considered, the value 1 is generally assigned to 00 but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context. <style data-mw-deduplicate="TemplateStyles:r1033199720">

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</style>For more details, see <a href="Zero_to_the_power_of_zero" title="Zero to the power of zero">Zero to the power of zero</a>.

<div class="mw-heading mw-heading3"><h3 id="Negative_exponents">Negative exponents</h3></div>

Exponentiation with negative exponents is defined by the following identity, which holds for any integer n and nonzero b:

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle b^{-n}={\frac {1}{b^{n}}}}</annotation>

</semantics>

</math><img src="./3bb09df7fe703ced09c5c135a5cb86c6e94ea77b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.645ex; height:5.343ex;" alt="{\displaystyle b^{-n}={\frac {1}{b^{n}}}}" loading="lazy">.<a href="#cite_note-:1-1">[1]</a></dd></dl>

Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \infty }</annotation>

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</math><img src="./c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy">).<a href="#cite_note-27">[26]</a>

This definition of exponentiation with negative exponents is the only one that allows extending the identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{m+n}=b^{m}\cdot b^{n}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle b^{m+n}=b^{m}\cdot b^{n}}</annotation>

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</math><img src="./259b25c515e091594d2f0ea2b75bf2628ce45f51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.604ex; height:2.509ex;" alt="{\displaystyle b^{m+n}=b^{m}\cdot b^{n}}" loading="lazy"> to negative exponents (consider the case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=-n}">

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<annotation encoding="application/x-tex">{\displaystyle m=-n}</annotation>

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</math><img src="./9d6e9afd09ef335f99a58a361faacfff3de778a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.342ex; height:2.176ex;" alt="{\displaystyle m=-n}" loading="lazy">).

The same definition applies to <a href="Invertible_element" class="mw-redirect" title="Invertible element">invertible elements</a> in a multiplicative <a href="Monoid" title="Monoid">monoid</a>, that is, an <a href="Algebraic_structure" title="Algebraic structure">algebraic structure</a>, with an associative multiplication and a <a href="Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">multiplicative identity</a> denoted 1 (for example, the <a href="Square_matrix" title="Square matrix">square matrices</a> of a given dimension). In particular, in such a structure, the inverse of an <a href="Invertible_element" class="mw-redirect" title="Invertible element">invertible element</a> x is standardly denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-1}.}">

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<annotation encoding="application/x-tex">{\displaystyle x^{-1}.}</annotation>

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</math><img src="./a9d686441acca0e660bc3343d83f1083adfb3dc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.309ex; height:2.676ex;" alt="{\displaystyle x^{-1}.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Identities_and_properties">Identities and properties</h3></div>

<div role="note" class="hatnote navigation-not-searchable">"Laws of Indices" redirects here. For the horse, see <a href="Laws_of_Indices_(horse)" title="Laws of Indices (horse)">Laws of Indices (horse)</a>.</div>

The following <a href="Identity_(mathematics)" title="Identity (mathematics)">identities</a>, often called <style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style>exponent rules, hold for all integer exponents, provided that the base is non-zero:<a href="#cite_note-:1-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}b^{m}\cdot b^{n}&=b^{m+n}\\\left(b^{m}\right)^{n}&=b^{m\cdot n}\\b^{n}\cdot c^{n}&=(b\cdot c)^{n}\end{aligned}}}</annotation>

</semantics>

</math><img src="./fb695430cd8efb8bd88498ab47063e2fd979b2f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.129ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}b^{m}\cdot b^{n}&=b^{m+n}\\\left(b^{m}\right)^{n}&=b^{m\cdot n}\\b^{n}\cdot c^{n}&=(b\cdot c)^{n}\end{aligned}}}" loading="lazy"></dd></dl>

Unlike addition and multiplication, exponentiation is not <a href="Commutative" class="mw-redirect" title="Commutative">commutative</a>: for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{3}=8}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 2^{3}=8}</annotation>

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</math><img src="./bb2dded8eba905e4a019b70abad935422b198db4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.478ex; height:2.676ex;" alt="{\displaystyle 2^{3}=8}" loading="lazy">, but reversing the operands gives the different value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{2}=9}">

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<annotation encoding="application/x-tex">{\displaystyle 3^{2}=9}</annotation>

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</math><img src="./6ce3bb0717e9aa3fd7c54d6676a7a7fe15e78e66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.478ex; height:2.676ex;" alt="{\displaystyle 3^{2}=9}" loading="lazy">. Also unlike addition and multiplication, exponentiation is not <a href="Associative" class="mw-redirect" title="Associative">associative</a>: for example, (23)2 = 82 = 64, whereas 2(32) = 29 = 512. Without parentheses, the conventional <a href="Order_of_operations" title="Order of operations">order of operations</a> for <a href="Serial_exponentiation" class="mw-redirect" title="Serial exponentiation">serial exponentiation</a> in superscript notation is top-down (or right-associative), not bottom-up<a href="#cite_note-Bronstein_1987-28">[27]</a><a href="#cite_note-NIST_2010-29">[28]</a><a href="#cite_note-Zeidler_2013-30">[29]</a> (or left-associative). That is,

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<annotation encoding="application/x-tex">{\displaystyle b^{p^{q}}=b^{\left(p^{q}\right)},}</annotation>

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</math><img src="./af2071cffd1ef446285db993b4a40cf57c29eba7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.695ex; height:3.176ex;" alt="{\displaystyle b^{p^{q}}=b^{\left(p^{q}\right)},}" loading="lazy"></dd></dl>

which, in general, is different from

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<annotation encoding="application/x-tex">{\displaystyle \left(b^{p}\right)^{q}=b^{pq}.}</annotation>

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</math><img src="./78911e3a5a6f894fdf2828cd6a7ad0db640d738d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.413ex; height:3.009ex;" alt="{\displaystyle \left(b^{p}\right)^{q}=b^{pq}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Powers_of_a_sum">Powers of a sum</h3></div>

The powers of a sum can normally be computed from the powers of the summands by the <a href="Binomial_formula" class="mw-redirect" title="Binomial formula">binomial formula</a>

<msup>

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</munderover>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle (a+b)^{n}=\sum _{i=0}^{n}{\binom {n}{i}}a^{i}b^{n-i}=\sum _{i=0}^{n}{\frac {n!}{i!(n-i)!}}a^{i}b^{n-i}.}</annotation>

</semantics>

</math><img src="./3d288a32615a5b91dc0060530c3df1721a6fa7ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.201ex; height:6.843ex;" alt="{\displaystyle (a+b)^{n}=\sum _{i=0}^{n}{\binom {n}{i}}a^{i}b^{n-i}=\sum _{i=0}^{n}{\frac {n!}{i!(n-i)!}}a^{i}b^{n-i}.}" loading="lazy"></dd></dl>

However, this formula is true only if the summands commute (i.e. that ab = ba), which is implied if they belong to a <a href="Algebraic_structure" title="Algebraic structure">structure</a> that is <a href="Commutative_property" title="Commutative property">commutative</a>. Otherwise, if a and b are, say, <a href="Square_matrix" title="Square matrix">square matrices</a> of the same size, this formula cannot be used. It follows that in <a href="Computer_algebra" title="Computer algebra">computer algebra</a>, many <a href="Algorithm" title="Algorithm">algorithms</a> involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose <a href="Computer_algebra_system" title="Computer algebra system">computer algebra systems</a> use a different notation (sometimes ^^ instead of ^) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.

<div class="mw-heading mw-heading3"><h3 id="Combinatorial_interpretation">Combinatorial interpretation</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="#Exponentiation_over_sets">Exponentiation over sets</a></div>

For nonnegative integers n and m, the value of nm is the number of <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> from a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of m elements to a set of n elements (see <a href="Cardinal_exponentiation" class="mw-redirect" title="Cardinal exponentiation">cardinal exponentiation</a>). Such functions can be represented as m-<a href="Tuple" title="Tuple">tuples</a> from an n-element set (or as m-letter words from an n-letter alphabet). Some examples for particular values of m and n are given in the following table:

<th>nm

</th>

<th>The nm possible m-tuples of elements from the set {1, ..., n}

</th></tr>

<tr>

<td>05 = 0

</td>

<td style="background: #EEE; color:black; vertical-align: middle; text-align: center;" class="table-cast">none

</td></tr>

<tr>

<td>14 = 1

</td>

<td>(1, 1, 1, 1)

</td></tr>

<tr>

<td>23 = 8

</td>

<td>(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)

</td></tr>

<tr>

<td>32 = 9

</td>

<td>(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)

</td></tr>

<tr>

<td>41 = 4

</td>

<td>(1), (2), (3), (4)

</td></tr>

<tr>

<td>50 = 1

</td>

<td>()

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Particular_bases">Particular bases</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Powers_of_ten">Powers of ten </h4></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Scientific_notation" title="Scientific notation">Scientific notation</a></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Power_of_10" title="Power of 10">Power of 10</a></div>

In the base ten (<a href="Decimal" title="Decimal">decimal</a>) number system, integer powers of 10 are written as the digit 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, 103 = 1000 and 10−4 = 0.0001.

Exponentiation with base <a href="10_(number)" class="mw-redirect" title="10 (number)">10</a> is used in <a href="Scientific_notation" title="Scientific notation">scientific notation</a> to denote large or small numbers. For instance, 299792458 m/s (the <a href="Speed_of_light" title="Speed of light">speed of light</a> in vacuum, in <a href="Metres_per_second" class="mw-redirect" title="Metres per second">metres per second</a>) can be written as 2.99792458×108 m/s and then <a href="Approximation" title="Approximation">approximated</a> as 2.998×108 m/s.

<a href="SI_prefix" class="mw-redirect" title="SI prefix">SI prefixes</a> based on powers of 10 are also used to describe small or large quantities. For example, the prefix <a href="Kilo-" title="Kilo-">kilo</a> means 103 = 1000, so a kilometre is 1000 m.

<div class="mw-heading mw-heading4"><h4 id="Powers_of_two">Powers of two</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Power_of_two" title="Power of two">Power of two</a></div>

The first negative powers of 2 have special names: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{-1}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 2^{-1}}</annotation>

</semantics>

</math><img src="./e4d1095a62660b99f5e9ef85ade17ab11a5d909f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.495ex; height:2.676ex;" alt="{\displaystyle 2^{-1}}" loading="lazy">is a <a href="One_half" title="One half">half</a>; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{-2}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 2^{-2}}</annotation>

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</math><img src="./4894401ca84c8f24cc9d94fa8aa516d1a1bc9497.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.495ex; height:2.676ex;" alt="{\displaystyle 2^{-2}}" loading="lazy"> is a <a href="4_(number)" class="mw-redirect" title="4 (number)">quarter</a>.

Powers of 2 appear in <a href="Set_theory" title="Set theory">set theory</a>, since a set with n members has a <a href="Power_set" title="Power set">power set</a>, the set of all of its <a href="Subset" title="Subset">subsets</a>, which has 2n members.

Integer powers of 2 are important in <a href="Computer_science" title="Computer science">computer science</a>. The positive integer powers 2n give the number of possible values for an n-<a href="Bit" title="Bit">bit</a> integer <a href="Binary_number" title="Binary number">binary number</a>; for example, a <a href="Byte" title="Byte">byte</a> may take 28 = 256 different values. The <a href="Binary_number_system" class="mw-redirect" title="Binary number system">binary number system</a> expresses any number as a sum of powers of 2, and denotes it as a sequence of 0 and 1, separated by a <a href="Binary_point" class="mw-redirect" title="Binary point">binary point</a>, where 1 indicates a power of 2 that appears in the sum; the exponent is determined by the place of this 1: the nonnegative exponents are the rank of the 1 on the left of the point (starting from 0), and the negative exponents are determined by the rank on the right of the point.

<div class="mw-heading mw-heading4"><h4 id="Powers_of_one">Powers of one</h4></div>

Every power of one equals: 1n = 1.

<div class="mw-heading mw-heading4"><h4 id="Powers_of_zero">Powers of zero</h4></div>

For a positive exponent n > 0, the nth power of zero is zero: 0n = 0. For a negative exponent, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{-n}=1/0^{n}=1/0}">

<msup>

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</msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle 0^{-n}=1/0^{n}=1/0}</annotation>

</semantics>

</math><img src="./55180bd85182dd6f0424b8b3c28882aa84389180.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.05ex; height:3.009ex;" alt="{\displaystyle 0^{-n}=1/0^{n}=1/0}" loading="lazy"> is undefined.

In some contexts (e.g., <a href="Combinatorics" title="Combinatorics">combinatorics</a>), the expression <a href="Zero_to_the_power_of_zero" title="Zero to the power of zero">00</a> is defined to be equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">; in others (e.g., <a href="Mathematical_analysis" title="Mathematical analysis">analysis</a>), it is often undefined.

<div class="mw-heading mw-heading4"><h4 id="Powers_of_negative_one">Powers of negative one</h4></div>

Since a negative number times another negative is positive, we have:<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{n}=\left\{{\begin{array}{rl}1&{\text{for even }}n,\\-1&{\text{for odd }}n.\\\end{array}}\right.}">

<msup>

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</msup>

<mrow>

<mtr>

<mtd>

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<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

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<mtd>

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</mtd>

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<annotation encoding="application/x-tex">{\displaystyle (-1)^{n}=\left\{{\begin{array}{rl}1&{\text{for even }}n,\\-1&{\text{for odd }}n.\\\end{array}}\right.}</annotation>

</semantics>

</math><img src="./cb2ae350f02efada680b04f1b7b3ec0b7828dde3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.458ex; height:6.176ex;" alt="{\displaystyle (-1)^{n}=\left\{{\begin{array}{rl}1&{\text{for even }}n,\\-1&{\text{for odd }}n.\\\end{array}}\right.}" loading="lazy"></blockquote>Because of this, powers of −1 are useful for expressing alternating <a href="Sequence" title="Sequence">sequences</a>. For a similar discussion of powers of the complex number i, see <a href="#nth_roots_of_a_complex_number">§ nth roots of a complex number</a>.

<div class="mw-heading mw-heading3"><h3 id="Large_exponents">Large exponents</h3></div>

The <a href="Limit_of_a_sequence" title="Limit of a sequence">limit of a sequence</a> of powers of a number greater than one diverges; in other words, the sequence grows without bound:

This can be read as "b to the power of n tends to <a href="Extended_real_number_line" title="Extended real number line">+∞</a> as n tends to infinity when b is greater than one".

Powers of a number with <a href="Absolute_value" title="Absolute value">absolute value</a> less than one tend to zero:

Any power of one is always one:

Powers of a negative number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\leq -1}">

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<annotation encoding="application/x-tex">{\displaystyle b\leq -1}</annotation>

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</math><img src="./f60e3adeea8aef80481c94cf32a5c0ac2f4a9a98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.067ex; height:2.343ex;" alt="{\displaystyle b\leq -1}" loading="lazy"> alternate between positive and negative as n alternates between even and odd, and thus do not tend to any limit as n grows.

If the exponentiated number varies while tending to 1 as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is

See <a href="#Exponential_function">§ Exponential function</a> below.

Other limits, in particular those of expressions that take on an <a href="Indeterminate_form" title="Indeterminate form">indeterminate form</a>, are described in <a href="#Limits_of_powers">§ Limits of powers</a> below.

<div class="mw-heading mw-heading3"><h3 id="Power_functions">Power functions</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Power_law" title="Power law">Power law</a></div>

Real functions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=cx^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)=cx^{n}}</annotation>

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</math><img src="./70728d15d74f24933b12f8f78bb044687603d933.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.071ex; height:2.843ex;" alt="{\displaystyle f(x)=cx^{n}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c\neq 0}</annotation>

</semantics>

</math><img src="./be27396bd0e62003728d08329a8767eee94409e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.268ex; height:2.676ex;" alt="{\displaystyle c\neq 0}" loading="lazy">, are sometimes called power functions.<a href="#cite_note-31">[30]</a> When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is an <a href="Integer" title="Integer">integer</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation>

</semantics>

</math><img src="./d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}" loading="lazy">, two primary families exist: for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> even, and for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> odd. In general for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c>0}</annotation>

</semantics>

</math><img src="./2ba126f626d61752f62eaacaf11761a54de4dc84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>0}" loading="lazy">, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is even <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=cx^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=cx^{n}}</annotation>

</semantics>

</math><img src="./70728d15d74f24933b12f8f78bb044687603d933.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.071ex; height:2.843ex;" alt="{\displaystyle f(x)=cx^{n}}" loading="lazy"> will tend towards positive <a href="Infinity_(mathematics)" class="mw-redirect" title="Infinity (mathematics)">infinity</a> with increasing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">, and also towards positive infinity with decreasing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. All graphs from the family of even power functions have the general shape of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=cx^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=cx^{2}}</annotation>

</semantics>

</math><img src="./e2a3b43230b3c215f0744a3e07a4c6e5b3d316e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.645ex; height:3.009ex;" alt="{\displaystyle y=cx^{2}}" loading="lazy">, flattening more in the middle as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> increases.<a href="#cite_note-Calculus:_Early_Transcendentals-32">[31]</a> Functions with this kind of <a href="Symmetry" title="Symmetry">symmetry</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(-x)=f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(-x)=f(x)}</annotation>

</semantics>

</math><img src="./185fd2e78903788bc5756b067d0ac6aae1846724.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.742ex; height:2.843ex;" alt="{\displaystyle f(-x)=f(x)}" loading="lazy">) are called <a href="Even_functions" class="mw-redirect" title="Even functions">even functions</a>.

When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is odd, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">'s <a href="Asymptotic" class="mw-redirect" title="Asymptotic">asymptotic</a> behavior reverses from positive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> to negative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle c>0}</annotation>

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</math><img src="./2ba126f626d61752f62eaacaf11761a54de4dc84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>0}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=cx^{n}}">

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(x)=cx^{n}}</annotation>

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</math><img src="./70728d15d74f24933b12f8f78bb044687603d933.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.071ex; height:2.843ex;" alt="{\displaystyle f(x)=cx^{n}}" loading="lazy"> will also tend towards positive <a href="Infinity_(mathematics)" class="mw-redirect" title="Infinity (mathematics)">infinity</a> with increasing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">, but towards negative infinity with decreasing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. All graphs from the family of odd power functions have the general shape of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=cx^{3}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=cx^{3}}</annotation>

</semantics>

</math><img src="./3ce21b52b7719cad5ae90c6a2f50a568e08b889e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.645ex; height:3.009ex;" alt="{\displaystyle y=cx^{3}}" loading="lazy">, flattening more in the middle as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> increases and losing all flatness there in the straight line for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=1}</annotation>

</semantics>

</math><img src="./d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy">. Functions with this kind of symmetry (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(-x)=-f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(-x)=-f(x)}</annotation>

</semantics>

</math><img src="./b022ffe516cf5bc26a68fd954753aa2bddf508f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.55ex; height:2.843ex;" alt="{\displaystyle f(-x)=-f(x)}" loading="lazy">) are called <a href="Odd_function" class="mw-redirect" title="Odd function">odd functions</a>.

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c<0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c<0}</annotation>

</semantics>

</math><img src="./26a48dc798956117afd8c429c39886678c0e7204.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c<0}" loading="lazy">, the opposite asymptotic behavior is true in each case.<a href="#cite_note-Calculus:_Early_Transcendentals-32">[31]</a>

<div class="mw-heading mw-heading3"><h3 id="Table_of_powers">Table of powers</h3></div>

<th><a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a> number

</th></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

</td></tr>

<tr>

<td>11

</td>

<td>121

</td>

<td>1331

</td>

<td>14641

</td>

<td>161051

</td>

<td>1771561

</td>

<td>19487171

</td>

<td>214358881

</td>

<td>2357947691

</td>

<td>25937424601

</td>

<td>285311670611

</td>

<td>

</td></tr>

<tr>

<td><a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a> number</td>

<td>

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Rational_exponents">Rational exponents</h2></div>

If x is a nonnegative <a href="Real_number" title="Real number">real number</a>, and n is a positive integer, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{1/n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{1/n}}</annotation>

</semantics>

</math><img src="./0acb961738e6ffb034db9b37250579f700c49d8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.192ex; height:2.843ex;" alt="{\displaystyle x^{1/n}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}}">

<mroot>

</mrow>

</mroot>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{x}}}</annotation>

</semantics>

</math><img src="./7b3ba2638d05cd9ed8dafae7e34986399e48ea99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{x}}}" loading="lazy"> denotes the unique nonnegative real <a href="Nth_root" title="Nth root">nth root</a> of x, that is, the unique nonnegative real number y such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{n}=x.}">

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<annotation encoding="application/x-tex">{\displaystyle y^{n}=x.}</annotation>

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</math><img src="./fcd015be3d9c359e2e372d2b97e8676b23eada1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.454ex; height:2.676ex;" alt="{\displaystyle y^{n}=x.}" loading="lazy">

If x is a positive real number, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p}{q}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {p}{q}}}</annotation>

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</math><img src="./e9903bc1de26879e5fc4c7f78b54b952bcbb800f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.006ex; height:5.343ex;" alt="{\displaystyle {\frac {p}{q}}}" loading="lazy"> is a <a href="Rational_number" title="Rational number">rational number</a>, with p and q > 0 integers, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x^{p/q}}">

<msup>

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<annotation encoding="application/x-tex">{\textstyle x^{p/q}}</annotation>

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</math><img src="./80ff84c2284e70bdb6d48563c84285add2479d49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.967ex; height:2.676ex;" alt="{\textstyle x^{p/q}}" loading="lazy"> is defined as

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle x^{\frac {p}{q}}=\left(x^{p}\right)^{\frac {1}{q}}=(x^{\frac {1}{q}})^{p}.}</annotation>

</semantics>

</math><img src="./d94c0d38a69c1fa2f6f8fda71e5937518f9edea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.781ex; height:4.009ex;" alt="{\displaystyle x^{\frac {p}{q}}=\left(x^{p}\right)^{\frac {1}{q}}=(x^{\frac {1}{q}})^{p}.}" loading="lazy"></dd></dl>

The equality on the right may be derived by setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x^{\frac {1}{q}},}">

<msup>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle y=x^{\frac {1}{q}},}</annotation>

</semantics>

</math><img src="./1829b7499fa70fe91a352b4159a1d042205e782a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.966ex; height:3.843ex;" alt="{\displaystyle y=x^{\frac {1}{q}},}" loading="lazy"> and writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{\frac {1}{q}})^{p}=y^{p}=\left((y^{p})^{q}\right)^{\frac {1}{q}}=\left((y^{q})^{p}\right)^{\frac {1}{q}}=(x^{p})^{\frac {1}{q}}.}">

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<mfrac>

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<mfrac>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle (x^{\frac {1}{q}})^{p}=y^{p}=\left((y^{p})^{q}\right)^{\frac {1}{q}}=\left((y^{q})^{p}\right)^{\frac {1}{q}}=(x^{p})^{\frac {1}{q}}.}</annotation>

</semantics>

</math><img src="./3e45d63134bef2bdd365f7b887db6c547d7ac595.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.253ex; height:4.009ex;" alt="{\displaystyle (x^{\frac {1}{q}})^{p}=y^{p}=\left((y^{p})^{q}\right)^{\frac {1}{q}}=\left((y^{q})^{p}\right)^{\frac {1}{q}}=(x^{p})^{\frac {1}{q}}.}" loading="lazy">

If r is a positive rational number, 0r = 0, by definition.

All these definitions are required for extending the identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{r})^{s}=x^{rs}}">

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<annotation encoding="application/x-tex">{\displaystyle (x^{r})^{s}=x^{rs}}</annotation>

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</math><img src="./ff34617bed2fde3e9522f90496115935dca81c38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.289ex; height:2.843ex;" alt="{\displaystyle (x^{r})^{s}=x^{rs}}" loading="lazy"> to rational exponents.

On the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers. For example, a negative real number has a real nth root, which is negative, if n is <a href="Odd_number" class="mw-redirect" title="Odd number">odd</a>, and no real root if n is even. In the latter case, whichever complex nth root one chooses for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\frac {1}{n}},}">

<msup>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle x^{\frac {1}{n}},}</annotation>

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</math><img src="./163cf3c2270119ede64f3b6a176b83a4b805451b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.846ex; height:3.676ex;" alt="{\displaystyle x^{\frac {1}{n}},}" loading="lazy"> the identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{a})^{b}=x^{ab}}">

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<annotation encoding="application/x-tex">{\displaystyle (x^{a})^{b}=x^{ab}}</annotation>

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</math><img src="./118169e9cd889db4377e54a9fd6aac8029db171a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.414ex; height:3.176ex;" alt="{\displaystyle (x^{a})^{b}=x^{ab}}" loading="lazy"> cannot be satisfied. For example,

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<annotation encoding="application/x-tex">{\displaystyle \left((-1)^{2}\right)^{\frac {1}{2}}=1^{\frac {1}{2}}=1\neq (-1)^{2\cdot {\frac {1}{2}}}=(-1)^{1}=-1.}</annotation>

</semantics>

</math><img src="./4a8ed578398d4131e26d4a50691e6e0d81447fe3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.499ex; height:4.509ex;" alt="{\displaystyle \left((-1)^{2}\right)^{\frac {1}{2}}=1^{\frac {1}{2}}=1\neq (-1)^{2\cdot {\frac {1}{2}}}=(-1)^{1}=-1.}" loading="lazy"></dd></dl>

See <a href="#Real_exponents">§ Real exponents</a> and <a href="#Non-integer_powers_of_complex_numbers">§ Non-integer powers of complex numbers</a> for details on the way these problems may be handled.

<div class="mw-heading mw-heading2"><h2 id="Real_exponents">Real exponents</h2></div>

For positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (<a href="#Limits_of_rational_exponents">§ Limits of rational exponents</a>, below), or in terms of the <a href="Logarithm" title="Logarithm">logarithm</a> of the base and the <a href="Exponential_function" title="Exponential function">exponential function</a> (<a href="#Powers_via_logarithms">§ Powers via logarithms</a>, below). The result is always a positive real number, and the <a href="#Identities_and_properties">identities and properties</a> shown above for integer exponents remain true with these definitions for real exponents. The second definition is more commonly used, since it generalizes straightforwardly to <a href="Complex_number" title="Complex number">complex</a> exponents.

On the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values. One may choose one of these values, called the <a href="Principal_value" title="Principal value">principal value</a>, but there is no choice of the principal value for which the identity

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<annotation encoding="application/x-tex">{\displaystyle \left(b^{r}\right)^{s}=b^{rs}}</annotation>

</semantics>

</math><img src="./88b9bb7d8cc1dca86605aee63b2c61373cd78cb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.625ex; height:3.009ex;" alt="{\displaystyle \left(b^{r}\right)^{s}=b^{rs}}" loading="lazy"></dd></dl>

is true; see <a href="#Failure_of_power_and_logarithm_identities">§ Failure of power and logarithm identities</a>. Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a <a href="Multivalued_function" title="Multivalued function">multivalued function</a>.

<div class="mw-heading mw-heading3"><h3 id="Limits_of_rational_exponents">Limits of rational exponents</h3></div>

Since any <a href="Irrational_number" title="Irrational number">irrational number</a> can be expressed as the <a href="Limit_of_a_sequence" title="Limit of a sequence">limit of a sequence</a> of rational numbers, exponentiation of a positive real number b with an arbitrary real exponent x can be defined by <a href="Continuous_function" title="Continuous function">continuity</a> with the rule<a href="#cite_note-Denlinger-33">[32]</a>

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<annotation encoding="application/x-tex">{\displaystyle b^{x}=\lim _{r(\in \mathbb {Q} )\to x}b^{r}\quad (b\in \mathbb {R} ^{+},\,x\in \mathbb {R} ),}</annotation>

</semantics>

</math><img src="./0d7c6e4e0c0e7fabb63678863e85ed3b8e413e2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.681ex; height:4.676ex;" alt="{\displaystyle b^{x}=\lim _{r(\in \mathbb {Q} )\to x}b^{r}\quad (b\in \mathbb {R} ^{+},\,x\in \mathbb {R} ),}" loading="lazy"></dd></dl>

where the limit is taken over rational values of r only. This limit exists for every positive b and every real x.

For example, if x = π, the <a href="Non-terminating_decimal" class="mw-redirect" title="Non-terminating decimal">non-terminating decimal</a> representation π = 3.14159... and the <a href="Monotone_function" class="mw-redirect" title="Monotone function">monotonicity</a> of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{\pi }:}">

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<annotation encoding="application/x-tex">{\displaystyle b^{\pi }:}</annotation>

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</math><img src="./8cf64434f23d05af3ac03eebd2cc07fa64c86a72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.464ex; height:2.343ex;" alt="{\displaystyle b^{\pi }:}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \left[b^{3},b^{4}\right],\left[b^{3.1},b^{3.2}\right],\left[b^{3.14},b^{3.15}\right],\left[b^{3.141},b^{3.142}\right],\left[b^{3.1415},b^{3.1416}\right],\left[b^{3.14159},b^{3.14160}\right],\ldots }</annotation>

</semantics>

</math><img src="./f09aec126692a109ddd2b003c78002512678e923.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:82.944ex; height:3.343ex;" alt="{\displaystyle \left[b^{3},b^{4}\right],\left[b^{3.1},b^{3.2}\right],\left[b^{3.14},b^{3.15}\right],\left[b^{3.141},b^{3.142}\right],\left[b^{3.1415},b^{3.1416}\right],\left[b^{3.14159},b^{3.14160}\right],\ldots }" loading="lazy"></dd></dl>

So, the upper bounds and the lower bounds of the intervals form two <a href="Sequence_(mathematics)" class="mw-redirect" title="Sequence (mathematics)">sequences</a> that have the same limit, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{\pi }.}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle b^{\pi }.}</annotation>

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</math><img src="./8d99a22002e3995dc8ed3c0cb26e5a1da4fbd2cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.819ex; height:2.343ex;" alt="{\displaystyle b^{\pi }.}" loading="lazy">

This defines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{x}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle b^{x}}</annotation>

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</math><img src="./2bb7406a338fb530330582bc63420d091897c709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.17ex; height:2.343ex;" alt="{\displaystyle b^{x}}" loading="lazy"> for every positive b and real x as a <a href="Continuous_function" title="Continuous function">continuous function</a> of b and x. See also <a href="Well-defined_expression" title="Well-defined expression">Well-defined expression</a>.<a href="#cite_note-34">[33]</a>

<div class="mw-heading mw-heading3"><h3 id="Exponential_function">Exponential function</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Exponential_function" title="Exponential function">Exponential function</a></div>

The exponential function may be defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto e^{x},}">

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<annotation encoding="application/x-tex">{\displaystyle x\mapsto e^{x},}</annotation>

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</math><img src="./f4abbbdd9b9faf016185bb1cd9d8ba9520ee0cc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.847ex; height:2.676ex;" alt="{\displaystyle x\mapsto e^{x},}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\approx 2.718}">

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<annotation encoding="application/x-tex">{\displaystyle e\approx 2.718}</annotation>

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</math><img src="./45e2bc9d17c0545d9f2792476c5473f296957270.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.479ex; height:2.176ex;" alt="{\displaystyle e\approx 2.718}" loading="lazy"> is <a href="Euler's_number" class="mw-redirect" title="Euler's number">Euler's number</a>, but to avoid <a href="Circular_reasoning" title="Circular reasoning">circular reasoning</a>, this definition cannot be used here. Rather, we give an independent definition of the exponential function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(x),}">

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<annotation encoding="application/x-tex">{\displaystyle \exp(x),}</annotation>

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</math><img src="./be4b828a01590da1a71cd22d30ffa9b09f093139.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.338ex; height:2.843ex;" alt="{\displaystyle \exp(x),}" loading="lazy"> and of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=\exp(1)}">

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<annotation encoding="application/x-tex">{\displaystyle e=\exp(1)}</annotation>

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</math><img src="./afa5c0edef9a0d9b60b8ece5fe43c2c706ea949f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.706ex; height:2.843ex;" alt="{\displaystyle e=\exp(1)}" loading="lazy">, relying only on positive integer powers (repeated multiplication). Then we sketch the proof that this agrees with the previous definition: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(x)=e^{x}.}">

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<annotation encoding="application/x-tex">{\displaystyle \exp(x)=e^{x}.}</annotation>

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</math><img src="./99130d96d40b7b53ae38e3a1f2ac5ee25d8215f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.693ex; height:2.843ex;" alt="{\displaystyle \exp(x)=e^{x}.}" loading="lazy">

There are <a href="Characterizations_of_the_exponential_function" title="Characterizations of the exponential function">many equivalent ways to define the exponential function</a>, one of them being

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<annotation encoding="application/x-tex">{\displaystyle \exp(x)=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}.}</annotation>

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</math><img src="./8255e5efd16a2e7e0334fcb31be93bfd4e091c9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.324ex; height:4.843ex;" alt="{\displaystyle \exp(x)=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}.}" loading="lazy"></dd></dl>

One has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(0)=1,}">

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<annotation encoding="application/x-tex">{\displaystyle \exp(0)=1,}</annotation>

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</math><img src="./f11d8e26383ece3acb8fa81df232091073759272.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.432ex; height:2.843ex;" alt="{\displaystyle \exp(0)=1,}" loading="lazy"> and the exponential identity (or multiplication rule) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(x)\exp(y)=\exp(x+y)}">

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<annotation encoding="application/x-tex">{\displaystyle \exp(x)\exp(y)=\exp(x+y)}</annotation>

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</math><img src="./58c25e33babee4e2c9d7a634900416f10ace4036.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.381ex; height:2.843ex;" alt="{\displaystyle \exp(x)\exp(y)=\exp(x+y)}" loading="lazy"> holds as well, since

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<annotation encoding="application/x-tex">{\displaystyle \exp(x)\exp(y)=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}\left(1+{\frac {y}{n}}\right)^{n}=\lim _{n\rightarrow \infty }\left(1+{\frac {x+y}{n}}+{\frac {xy}{n^{2}}}\right)^{n},}</annotation>

</semantics>

</math><img src="./3db8e381f2e32b911f814cee79fab417180cba84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:71.18ex; height:6.176ex;" alt="{\displaystyle \exp(x)\exp(y)=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}\left(1+{\frac {y}{n}}\right)^{n}=\lim _{n\rightarrow \infty }\left(1+{\frac {x+y}{n}}+{\frac {xy}{n^{2}}}\right)^{n},}" loading="lazy"></dd></dl>

and the second-order term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {xy}{n^{2}}}}">

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {xy}{n^{2}}}}</annotation>

</semantics>

</math><img src="./a07221ca1ab194f3c09b55648a668efc2db8908a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:3.321ex; height:5.176ex;" alt="{\displaystyle {\frac {xy}{n^{2}}}}" loading="lazy"> does not affect the limit, yielding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(x)\exp(y)=\exp(x+y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp(x)\exp(y)=\exp(x+y)}</annotation>

</semantics>

</math><img src="./58c25e33babee4e2c9d7a634900416f10ace4036.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.381ex; height:2.843ex;" alt="{\displaystyle \exp(x)\exp(y)=\exp(x+y)}" loading="lazy">.

Euler's number can be defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=\exp(1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e=\exp(1)}</annotation>

</semantics>

</math><img src="./afa5c0edef9a0d9b60b8ece5fe43c2c706ea949f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.706ex; height:2.843ex;" alt="{\displaystyle e=\exp(1)}" loading="lazy">. It follows from the preceding equations that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(x)=e^{x}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp(x)=e^{x}}</annotation>

</semantics>

</math><img src="./13cc0e0007b0b0baf6553e5cd4ea883f030cc03b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.046ex; height:2.843ex;" alt="{\displaystyle \exp(x)=e^{x}}" loading="lazy"> when x is an integer (this results from the repeated-multiplication definition of the exponentiation). If x is real, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(x)=e^{x}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp(x)=e^{x}}</annotation>

</semantics>

</math><img src="./13cc0e0007b0b0baf6553e5cd4ea883f030cc03b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.046ex; height:2.843ex;" alt="{\displaystyle \exp(x)=e^{x}}" loading="lazy"> results from the definitions given in preceding sections, by using the exponential identity if x is rational, and the continuity of the exponential function otherwise.

The limit that defines the exponential function converges for every <a href="Complex_number" title="Complex number">complex</a> value of x, and therefore it can be used to extend the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp(z)}</annotation>

</semantics>

</math><img src="./fc14dcff41f956163726d55369923012da262b3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.45ex; height:2.843ex;" alt="{\displaystyle \exp(z)}" loading="lazy">, and thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{z},}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{z},}</annotation>

</semantics>

</math><img src="./896eae8a27bbed531da8eccafe42f5c73f9f2687.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle e^{z},}" loading="lazy"> from the real numbers to any complex argument z. This extended exponential function still satisfies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.

<div class="mw-heading mw-heading3"><h3 id="Powers_via_logarithms">Powers via logarithms</h3></div>

The definition of ex as the exponential function allows defining bx for every positive real numbers b, in terms of exponential and <a href="Logarithm" title="Logarithm">logarithm</a> function. Specifically, the fact that the <a href="Natural_logarithm" title="Natural logarithm">natural logarithm</a> ln(x) is the <a href="Inverse_function" title="Inverse function">inverse</a> of the exponential function ex means that one has

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=\exp(\ln b)=e^{\ln b}}</annotation>

</semantics>

</math><img src="./7dd379a831e1ea8d4431e21eceefdc6e48455d03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.66ex; height:3.176ex;" alt="{\displaystyle b=\exp(\ln b)=e^{\ln b}}" loading="lazy"></dd></dl>

for every b > 0. For preserving the identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e^{x})^{y}=e^{xy},}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (e^{x})^{y}=e^{xy},}</annotation>

</semantics>

</math><img src="./6e9d415c7df0fd4d6aa6c939f16ad611ad93bd97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.933ex; height:2.843ex;" alt="{\displaystyle (e^{x})^{y}=e^{xy},}" loading="lazy"> one must have

<msup>

</mrow>

</msup>

<msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b^{x}=\left(e^{\ln b}\right)^{x}=e^{x\ln b}}</annotation>

</semantics>

</math><img src="./f586f25b09508ce982f08003e100228437bd2e8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.556ex; height:3.509ex;" alt="{\displaystyle b^{x}=\left(e^{\ln b}\right)^{x}=e^{x\ln b}}" loading="lazy"></dd></dl>

So, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x\ln b}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{x\ln b}}</annotation>

</semantics>

</math><img src="./62ac636417011acc5f18ab9052159e2b7461429f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.107ex; height:2.676ex;" alt="{\displaystyle e^{x\ln b}}" loading="lazy"> can be used as an alternative definition of bx for any positive real b. This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.

<div class="mw-heading mw-heading2"><h2 id="Complex_exponents_with_a_positive_real_base">Complex exponents with a positive real base</h2></div>

If b is a positive real number, exponentiation with base b and <a href="Complex_number" title="Complex number">complex</a> exponent z is defined by means of the exponential function with complex argument (see the end of <a href="#Exponential_function">§ Exponential function</a>, above) as

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b^{z}=e^{(z\ln b)},}</annotation>

</semantics>

</math><img src="./b2782abb56e74b86a3885845cfea6b948cc4754f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.96ex; height:3.176ex;" alt="{\displaystyle b^{z}=e^{(z\ln b)},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ln b}</annotation>

</semantics>

</math><img src="./8414fc5ee97aafb1ba8785fbd54f49b5d16af86b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.324ex; height:2.176ex;" alt="{\displaystyle \ln b}" loading="lazy"> denotes the <a href="Natural_logarithm" title="Natural logarithm">natural logarithm</a> of b.

This satisfies the identity

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b^{z+t}=b^{z}b^{t},}</annotation>

</semantics>

</math><img src="./c57616b4e23e2f386916628c6d3e01fed886049e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.44ex; height:2.843ex;" alt="{\displaystyle b^{z+t}=b^{z}b^{t},}" loading="lazy"></dd></dl>

In general,

<msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \left(b^{z}\right)^{t}}</annotation>

</semantics>

</math><img src="./66ed121c6e71b137704ec3a2eea06f8e9d368db0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.634ex; height:3.176ex;" alt="{\textstyle \left(b^{z}\right)^{t}}" loading="lazy"> is not defined, since bz is not a real number. If a meaning is given to the exponentiation of a complex number (see <a href="#Non-integer_powers_of_complex_numbers">§ Non-integer powers of complex numbers</a>, below), one has, in general,

<msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(b^{z}\right)^{t}\neq b^{zt},}</annotation>

</semantics>

</math><img src="./24e7146d16aacce800bb8ecbe9141a66fefa1e55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.973ex; height:3.176ex;" alt="{\displaystyle \left(b^{z}\right)^{t}\neq b^{zt},}" loading="lazy"></dd></dl>

unless z is real or t is an integer.

<a href="Euler's_formula" title="Euler's formula">Euler's formula</a>,

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{iy}=\cos y+i\sin y,}</annotation>

</semantics>

</math><img src="./3734533e05fd60929400f4ecce01fe846b9fae03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.527ex; height:3.009ex;" alt="{\displaystyle e^{iy}=\cos y+i\sin y,}" loading="lazy"></dd></dl>

allows expressing the <a href="Polar_form" class="mw-redirect" title="Polar form">polar form</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{z}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b^{z}}</annotation>

</semantics>

</math><img src="./4adc69dc65f48bb53260814c2b4680760b28099f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.999ex; height:2.343ex;" alt="{\displaystyle b^{z}}" loading="lazy"> in terms of the <a href="Real_and_imaginary_parts" class="mw-redirect" title="Real and imaginary parts">real and imaginary parts</a> of z, namely

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b^{x+iy}=b^{x}(\cos(y\ln b)+i\sin(y\ln b)),}</annotation>

</semantics>

</math><img src="./583ed52441423846be9dc329680fd2578383defe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.906ex; height:3.176ex;" alt="{\displaystyle b^{x+iy}=b^{x}(\cos(y\ln b)+i\sin(y\ln b)),}" loading="lazy"></dd></dl>

where the <a href="Absolute_value" title="Absolute value">absolute value</a> of the <a href="Trigonometry" title="Trigonometry">trigonometric</a> factor is one. This results from

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b^{x+iy}=b^{x}b^{iy}=b^{x}e^{iy\ln b}=b^{x}(\cos(y\ln b)+i\sin(y\ln b)).}</annotation>

</semantics>

</math><img src="./08bc068f38c3b8aa1f519190d6b3e4d6c09b44fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.609ex; height:3.176ex;" alt="{\displaystyle b^{x+iy}=b^{x}b^{iy}=b^{x}e^{iy\ln b}=b^{x}(\cos(y\ln b)+i\sin(y\ln b)).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Non-integer_exponents_with_a_complex_base">Non-integer exponents with a complex base</h2></div>

In the preceding sections, exponentiation with non-integer exponents has been defined for positive real bases only. For other bases, difficulties appear already with the apparently simple case of nth roots, that is, of exponents <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/n,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/n,}</annotation>

</semantics>

</math><img src="./de8b6b7358e7a75273d3a0612d1ef0dcedb77730.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.366ex; height:2.843ex;" alt="{\displaystyle 1/n,}" loading="lazy"> where n is a positive integer. Although the general theory of exponentiation with non-integer exponents applies to nth roots, this case deserves to be considered first, since it does not need to use <a href="Complex_logarithm" title="Complex logarithm">complex logarithms</a>, and is therefore easier to understand.

<div class="mw-heading mw-heading3"><h3 id="nth_roots_of_a_complex_number">nth roots of a complex number</h3></div>

Every nonzero complex number z may be written in <a href="Polar_form" class="mw-redirect" title="Polar form">polar form</a> as

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=\rho e^{i\theta }=\rho (\cos \theta +i\sin \theta ),}</annotation>

</semantics>

</math><img src="./d31d7613174f2b2478b44567c33b4f16473ca26b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.751ex; height:3.176ex;" alt="{\displaystyle z=\rho e^{i\theta }=\rho (\cos \theta +i\sin \theta ),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> is the <a href="Absolute_value" title="Absolute value">absolute value</a> of z, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> is its <a href="Argument_(complex_analysis)" title="Argument (complex analysis)">argument</a>. The argument is defined <a href="Up_to" title="Up to">up to</a> an integer multiple of 2π; this means that, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> is the argument of a complex number, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta +2k\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta +2k\pi }</annotation>

</semantics>

</math><img src="./d0a1587f2126377070c97b9978f744a8c7c3287b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.637ex; height:2.343ex;" alt="{\displaystyle \theta +2k\pi }" loading="lazy"> is also an argument of the same complex number for every integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">.

The polar form of the product of two complex numbers is obtained by multiplying the absolute values and adding the arguments. It follows that the polar form of an nth root of a complex number can be obtained by taking the nth root of the absolute value and dividing its argument by n:

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msup>

<mroot>

</mrow>

</mroot>

</mrow>

<msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(\rho e^{i\theta }\right)^{\frac {1}{n}}={\sqrt[{n}]{\rho }}\,e^{\frac {i\theta }{n}}.}</annotation>

</semantics>

</math><img src="./1fc4ae73419a2cabb0133293dc1bfa19d743b1d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.364ex; height:4.676ex;" alt="{\displaystyle \left(\rho e^{i\theta }\right)^{\frac {1}{n}}={\sqrt[{n}]{\rho }}\,e^{\frac {i\theta }{n}}.}" loading="lazy"></dd></dl>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

</semantics>

</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy"> is added to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy">, the complex number is not changed, but this adds <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2i\pi /n}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2i\pi /n}</annotation>

</semantics>

</math><img src="./436fb55c4a5761ef1485bd42294a98a9dd40dd37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.854ex; height:2.843ex;" alt="{\displaystyle 2i\pi /n}" loading="lazy"> to the argument of the nth root, and provides a new nth root. This can be done n times (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0,1,...,n-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=0,1,...,n-1}</annotation>

</semantics>

</math><img src="./f0e1779e0578a7b355867d896b466d6eaf2c6108.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.236ex; height:2.509ex;" alt="{\displaystyle k=0,1,...,n-1}" loading="lazy">), and provides the n nth roots of the complex number:

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \left(\rho e^{i(\theta +2k\pi )}\right)^{\frac {1}{n}}={\sqrt[{n}]{\rho }}\,e^{\frac {i(\theta +2k\pi )}{n}}.}</annotation>

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</math><img src="./b107d807fae0ecce0467f0f1812e2368a38f1ea8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.392ex; height:5.843ex;" alt="{\displaystyle \left(\rho e^{i(\theta +2k\pi )}\right)^{\frac {1}{n}}={\sqrt[{n}]{\rho }}\,e^{\frac {i(\theta +2k\pi )}{n}}.}" loading="lazy"></dd></dl>

It is usual to choose one of the n nth root as the <a href="Principal_root" class="mw-redirect" title="Principal root">principal root</a>. The common choice is to choose the nth root for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi <\theta \leq \pi ,}">

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<annotation encoding="application/x-tex">{\displaystyle -\pi <\theta \leq \pi ,}</annotation>

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</math><img src="./622f6ef7b35fe49aa9a1854baec3c208a5b2200d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.406ex; height:2.509ex;" alt="{\displaystyle -\pi <\theta \leq \pi ,}" loading="lazy"> that is, the nth root that has the largest real part, and, if there are two, the one with positive imaginary part. This makes the principal nth root a <a href="Continuous_function" title="Continuous function">continuous function</a> in the whole complex plane, except for negative real values of the <a href="Radicand" class="mw-redirect" title="Radicand">radicand</a>. This function equals the usual nth root for positive real radicands. For negative real radicands, and odd exponents, the principal nth root is not real, although the usual nth root is real. <a href="Analytic_continuation" title="Analytic continuation">Analytic continuation</a> shows that the principal nth root is the unique <a href="Complex_differentiable" class="mw-redirect" title="Complex differentiable">complex differentiable</a> function that extends the usual nth root to the complex plane without the nonpositive real numbers.

If the complex number is moved around zero by increasing its argument, after an increment of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi ,}">

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<annotation encoding="application/x-tex">{\displaystyle 2\pi ,}</annotation>

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</math><img src="./461d52fa6fb5a16ef8a24e871488584db5398489.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.141ex; height:2.509ex;" alt="{\displaystyle 2\pi ,}" loading="lazy"> the complex number comes back to its initial position, and its nth roots are <a href="Circular_permutation" class="mw-redirect" title="Circular permutation">permuted circularly</a> (they are multiplied by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="e^{2i\pi /n}">

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</math><img src="./3e23f0bc2c511a265f3b2a6e01bb6462c044fad7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.455ex; height:2.843ex;" alt="e^{2i\pi /n}" loading="lazy">). This shows that it is not possible to define a nth root function that is continuous in the whole complex plane.

<div class="mw-heading mw-heading4"><h4 id="Roots_of_unity">Roots of unity</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Root_of_unity" title="Root of unity">Root of unity</a></div>

The nth roots of unity are the n complex numbers such that wn = 1, where n is a positive integer. They arise in various areas of mathematics, such as in <a href="Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a> or algebraic solutions of algebraic equations (<a href="Lagrange_resolvent" class="mw-redirect" title="Lagrange resolvent">Lagrange resolvent</a>).

The n nth roots of unity are the n first powers of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =e^{\frac {2\pi i}{n}}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega =e^{\frac {2\pi i}{n}}}</annotation>

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</math><img src="./93780c907ae1a68d17d1b856d4055b86011e6703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.589ex; height:3.343ex;" alt="{\displaystyle \omega =e^{\frac {2\pi i}{n}}}" loading="lazy">, that is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=\omega ^{0}=\omega ^{n},\omega =\omega ^{1},\omega ^{2},...,\omega ^{n-1}.}">

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<annotation encoding="application/x-tex">{\displaystyle 1=\omega ^{0}=\omega ^{n},\omega =\omega ^{1},\omega ^{2},...,\omega ^{n-1}.}</annotation>

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</math><img src="./a929b54e7187413f31e34e88d19c5c45ed0532a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.717ex; height:3.009ex;" alt="{\displaystyle 1=\omega ^{0}=\omega ^{n},\omega =\omega ^{1},\omega ^{2},...,\omega ^{n-1}.}" loading="lazy"> The nth roots of unity that have this generating property are called primitive nth roots of unity; they have the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{k}=e^{\frac {2k\pi i}{n}},}">

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<annotation encoding="application/x-tex">{\displaystyle \omega ^{k}=e^{\frac {2k\pi i}{n}},}</annotation>

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</math><img src="./c90ccb18ece08f3b4cd4357d81bbbc328c8d1499.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.02ex; height:3.676ex;" alt="{\displaystyle \omega ^{k}=e^{\frac {2k\pi i}{n}},}" loading="lazy"> with k <a href="Coprime_integers" title="Coprime integers">coprime</a> with n. The unique primitive square root of unity is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1;}">

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<annotation encoding="application/x-tex">{\displaystyle -1;}</annotation>

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</math><img src="./a8b7acf3b7827f0f6abb10a7b09d13d47ffaa3ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.617ex; height:2.509ex;" alt="{\displaystyle -1;}" loading="lazy"> the primitive fourth roots of unity are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i.}">

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<annotation encoding="application/x-tex">{\displaystyle -i.}</annotation>

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</math><img src="./b2712f3f03e79f9422bcba945d28eb4f6701356f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.257ex; height:2.343ex;" alt="{\displaystyle -i.}" loading="lazy">

The nth roots of unity allow expressing all nth roots of a complex number z as the n products of a given nth roots of z with a nth root of unity.

Geometrically, the nth roots of unity lie on the <a href="Unit_circle" title="Unit circle">unit circle</a> of the <a href="Complex_plane" title="Complex plane">complex plane</a> at the vertices of a <a href="Regular_polygon" title="Regular polygon">regular n-gon</a> with one vertex on the real number 1.

As the number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\frac {2k\pi i}{n}}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{\frac {2k\pi i}{n}}}</annotation>

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</math><img src="./bf0bcff6f3fae7b9951092a0f33d31d56a925299.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.74ex; height:3.343ex;" alt="{\displaystyle e^{\frac {2k\pi i}{n}}}" loading="lazy"> is the primitive nth root of unity with the smallest positive <a href="Argument_(complex_analysis)" title="Argument (complex analysis)">argument</a>, it is called the principal primitive nth root of unity, sometimes shortened as principal nth root of unity, although this terminology can be confused with the <a href="Principal_value" title="Principal value">principal value</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{1/n}}">

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<annotation encoding="application/x-tex">{\displaystyle 1^{1/n}}</annotation>

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</math><img src="./4bb7a733f35424aaf85d61b6d8e0a72104ceecda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.025ex; height:2.843ex;" alt="{\displaystyle 1^{1/n}}" loading="lazy">, which is 1.<a href="#cite_note-35">[34]</a><a href="#cite_note-36">[35]</a><a href="#cite_note-37">[36]</a>

<div class="mw-heading mw-heading3"><h3 id="Complex_exponentiation">Complex exponentiation</h3></div>

Defining exponentiation with complex bases leads to difficulties that are similar to those described in the preceding section, except that there are, in general, infinitely many possible values for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z^{w}">

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</math><img src="./3ee32b40d1daf34c11caca8a8ed8504b1d8b5c85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.343ex;" alt="z^{w}" loading="lazy">. So, either a <a href="Principal_value" title="Principal value">principal value</a> is defined, which is not continuous for the values of z that are real and nonpositive, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z^{w}">

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</math><img src="./3ee32b40d1daf34c11caca8a8ed8504b1d8b5c85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.343ex;" alt="z^{w}" loading="lazy"> is defined as a <a href="Multivalued_function" title="Multivalued function">multivalued function</a>.

In all cases, the <a href="Complex_logarithm" title="Complex logarithm">complex logarithm</a> is used to define complex exponentiation as

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<annotation encoding="application/x-tex">{\displaystyle z^{w}=e^{w\log z},}</annotation>

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</math><img src="./919b1939ec06cd797ef8f5416f0bdab75c3fb3e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.382ex; height:3.009ex;" alt="{\displaystyle z^{w}=e^{w\log z},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log z}">

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<annotation encoding="application/x-tex">{\displaystyle \log z}</annotation>

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</math><img src="./3d208820c31babc9dfabefde37b4309cfceecc6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.447ex; height:2.509ex;" alt="{\displaystyle \log z}" loading="lazy"> is the variant of the complex logarithm that is used, which is a function or a <a href="Multivalued_function" title="Multivalued function">multivalued function</a> such that

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<annotation encoding="application/x-tex">{\displaystyle e^{\log z}=z}</annotation>

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</math><img src="./9bf29a5bea3613b50b27bb5d7d638fa7f08da4e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.76ex; height:2.676ex;" alt="{\displaystyle e^{\log z}=z}" loading="lazy"></dd></dl>

for every z in its <a href="Domain_of_a_function" title="Domain of a function">domain of definition</a>.

<div class="mw-heading mw-heading4"><h4 id="Principal_value">Principal value</h4></div>

The <a href="Principal_value" title="Principal value">principal value</a> of the <a href="Complex_logarithm" title="Complex logarithm">complex logarithm</a> is the unique continuous function, commonly denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log ,}">

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<annotation encoding="application/x-tex">{\displaystyle \log ,}</annotation>

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</math><img src="./35741b181e11f6da346fc21f89ec71c566bce3d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.619ex; height:2.509ex;" alt="{\displaystyle \log ,}" loading="lazy"> such that, for every nonzero complex number z,

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<annotation encoding="application/x-tex">{\displaystyle e^{\log z}=z,}</annotation>

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</math><img src="./f8e5c8de444735fddd4d4eff764cc770ab940239.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.407ex; height:3.009ex;" alt="{\displaystyle e^{\log z}=z,}" loading="lazy"></dd></dl>

and the <a href="Argument_(complex_analysis)" title="Argument (complex analysis)">argument</a> of z satisfies

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<annotation encoding="application/x-tex">{\displaystyle -\pi <\operatorname {Arg} z\leq \pi .}</annotation>

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</math><img src="./fac05203410480daa575b0d482d8707d20487fb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.608ex; height:2.509ex;" alt="{\displaystyle -\pi <\operatorname {Arg} z\leq \pi .}" loading="lazy"></dd></dl>

The principal value of the complex logarithm is not defined for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0,}">

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<annotation encoding="application/x-tex">{\displaystyle z=0,}</annotation>

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</math><img src="./463359fa7c7563dc29f2079e63195b0035f1ab5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.996ex; height:2.509ex;" alt="{\displaystyle z=0,}" loading="lazy"> it is <a href="Continuous_function" title="Continuous function">discontinuous</a> at negative real values of z, and it is <a href="Holomorphic" class="mw-redirect" title="Holomorphic">holomorphic</a> (that is, complex differentiable) elsewhere. If z is real and positive, the principal value of the complex logarithm is the natural logarithm: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log z=\ln z.}">

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<annotation encoding="application/x-tex">{\displaystyle \log z=\ln z.}</annotation>

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</math><img src="./aa06744c88ccc1b658f6ab4f102c5dd7bda78af6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.607ex; height:2.509ex;" alt="{\displaystyle \log z=\ln z.}" loading="lazy">

The principal value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{w}}">

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</math><img src="./3010585a40a0ee9155d34b74b5029fd163197dce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.343ex;" alt="{\displaystyle z^{w}}" loading="lazy"> is defined as

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<annotation encoding="application/x-tex">{\displaystyle z^{w}=e^{w\log z},}</annotation>

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</math><img src="./919b1939ec06cd797ef8f5416f0bdab75c3fb3e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.382ex; height:3.009ex;" alt="{\displaystyle z^{w}=e^{w\log z},}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log z}">

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<annotation encoding="application/x-tex">{\displaystyle \log z}</annotation>

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</math><img src="./3d208820c31babc9dfabefde37b4309cfceecc6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.447ex; height:2.509ex;" alt="{\displaystyle \log z}" loading="lazy"> is the principal value of the logarithm.

The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (z,w)\to z^{w}}">

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</math><img src="./f4950e15e46695f3a19b636af4a9869a89086485.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.709ex; height:2.843ex;" alt="{\displaystyle (z,w)\to z^{w}}" loading="lazy"> is holomorphic except in the neighbourhood of the points where z is real and nonpositive.

If z is real and positive, the principal value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{w}}">

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</math><img src="./3010585a40a0ee9155d34b74b5029fd163197dce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.343ex;" alt="{\displaystyle z^{w}}" loading="lazy"> equals its usual value defined above. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=1/n,}">

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<annotation encoding="application/x-tex">{\displaystyle w=1/n,}</annotation>

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</math><img src="./48c86cd5d9f8943264c1e1464894339f32a69953.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.129ex; height:2.843ex;" alt="{\displaystyle w=1/n,}" loading="lazy"> where n is an integer, this principal value is the same as the one defined above.

<div class="mw-heading mw-heading4"><h4 id="Multivalued_function">Multivalued function</h4></div>

In some contexts, there is a problem with the discontinuity of the principal values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log z}">

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</math><img src="./3d208820c31babc9dfabefde37b4309cfceecc6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.447ex; height:2.509ex;" alt="{\displaystyle \log z}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{w}}">

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<annotation encoding="application/x-tex">{\displaystyle z^{w}}</annotation>

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</math><img src="./3010585a40a0ee9155d34b74b5029fd163197dce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.343ex;" alt="{\displaystyle z^{w}}" loading="lazy"> at the negative real values of z. In this case, it is useful to consider these functions as <a href="Multivalued_function" title="Multivalued function">multivalued functions</a>.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log z}">

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</math><img src="./3d208820c31babc9dfabefde37b4309cfceecc6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.447ex; height:2.509ex;" alt="{\displaystyle \log z}" loading="lazy"> denotes one of the values of the multivalued logarithm (typically its principal value), the other values are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2ik\pi +\log z,}">

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</math><img src="./a0ba576c110a76b4e6c2447bc68d9f549536b031.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.442ex; height:2.509ex;" alt="{\displaystyle 2ik\pi +\log z,}" loading="lazy"> where k is any integer. Similarly, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{w}}">

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<annotation encoding="application/x-tex">{\displaystyle z^{w}}</annotation>

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</math><img src="./3010585a40a0ee9155d34b74b5029fd163197dce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.343ex;" alt="{\displaystyle z^{w}}" loading="lazy"> is one value of the exponentiation, then the other values are given by

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<annotation encoding="application/x-tex">{\displaystyle e^{w(2ik\pi +\log z)}=z^{w}e^{2ik\pi w},}</annotation>

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</math><img src="./ca78d9b8d1931238d0d17e03cb7ef7c2b9f548a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.421ex; height:3.176ex;" alt="{\displaystyle e^{w(2ik\pi +\log z)}=z^{w}e^{2ik\pi w},}" loading="lazy"></dd></dl>

where k is any integer.

Different values of k give different values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{w}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{w}}</annotation>

</semantics>

</math><img src="./3010585a40a0ee9155d34b74b5029fd163197dce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.343ex;" alt="{\displaystyle z^{w}}" loading="lazy"> unless w is a <a href="Rational_number" title="Rational number">rational number</a>, that is, there is an integer d such that dw is an integer. This results from the <a href="Periodic_function" title="Periodic function">periodicity</a> of the exponential function, more specifically, that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{a}=e^{b}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{a}=e^{b}}</annotation>

</semantics>

</math><img src="./d3d4b37843c43ead2580e8b843b530bda2f02a75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.305ex; height:2.676ex;" alt="{\displaystyle e^{a}=e^{b}}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a-b}</annotation>

</semantics>

</math><img src="./1b80866c2bf2f1bc1f2e4c97e7937f5663150ea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a-b}" loading="lazy"> is an integer multiple of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi i.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi i.}</annotation>

</semantics>

</math><img src="./5df66e242f62d9f27aa62c27256ce715ed91c0e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.944ex; height:2.176ex;" alt="{\displaystyle 2\pi i.}" loading="lazy">

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w={\frac {m}{n}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w={\frac {m}{n}}}</annotation>

</semantics>

</math><img src="./57a0ffc91b6043537ee495e83f2e3c5b7f022208.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.639ex; height:4.676ex;" alt="{\displaystyle w={\frac {m}{n}}}" loading="lazy"> is a rational number with m and n <a href="Coprime_integers" title="Coprime integers">coprime integers</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n>0,}</annotation>

</semantics>

</math><img src="./f2b90cfe174f0cf1c285539df4d03d339af13d87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.302ex; height:2.509ex;" alt="{\displaystyle n>0,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{w}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{w}}</annotation>

</semantics>

</math><img src="./3010585a40a0ee9155d34b74b5029fd163197dce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.343ex;" alt="{\displaystyle z^{w}}" loading="lazy"> has exactly n values. In the case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=1,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m=1,}</annotation>

</semantics>

</math><img src="./ef9b2e0c3a281b794d3547a628fc6796720f601c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.948ex; height:2.509ex;" alt="{\displaystyle m=1,}" loading="lazy"> these values are the same as those described in <a href="#nth_roots_of_a_complex_number">§ nth roots of a complex number</a>. If w is an integer, there is only one value that agrees with that of <a href="#Integer_exponents">§ Integer exponents</a>.

The multivalued exponentiation is holomorphic for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\neq 0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\neq 0,}</annotation>

</semantics>

</math><img src="./7708b7928735bf6c760aeb3e8cc07e50ae1495c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.996ex; height:2.676ex;" alt="{\displaystyle z\neq 0,}" loading="lazy"> in the sense that its <a href="Graph_of_a_function" title="Graph of a function">graph</a> consists of several sheets that define each a holomorphic function in the neighborhood of every point. If z varies continuously along a circle around 0, then, after a turn, the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{w}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{w}}</annotation>

</semantics>

</math><img src="./3010585a40a0ee9155d34b74b5029fd163197dce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.343ex;" alt="{\displaystyle z^{w}}" loading="lazy"> has changed of sheet.

<div class="mw-heading mw-heading4"><h4 id="Computation">Computation</h4></div>

The canonical form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+iy}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x+iy}</annotation>

</semantics>

</math><img src="./ceb1c6ce62a20dbfe9cb3d82dca889577b469703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.128ex; height:2.509ex;" alt="{\displaystyle x+iy}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{w}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{w}}</annotation>

</semantics>

</math><img src="./3010585a40a0ee9155d34b74b5029fd163197dce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.343ex;" alt="{\displaystyle z^{w}}" loading="lazy"> can be computed from the canonical form of z and w. Although this can be described by a single formula, it is clearer to split the computation in several steps.

<ul><li><a href="Polar_form" class="mw-redirect" title="Polar form">Polar form</a> of z. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+ib}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=a+ib}</annotation>

</semantics>

</math><img src="./768804bf1e5f2b40378497f93bb04a10ce64098b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.057ex; height:2.343ex;" alt="{\displaystyle z=a+ib}" loading="lazy"> is the canonical form of z (a and b being real), then its polar form is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\rho e^{i\theta }=\rho (\cos \theta +i\sin \theta ),}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=\rho e^{i\theta }=\rho (\cos \theta +i\sin \theta ),}</annotation>

</semantics>

</math> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \rho ={\sqrt {a^{2}+b^{2}}}}">

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \rho ={\sqrt {a^{2}+b^{2}}}}</annotation>

</semantics>

</math><img src="./1119907627e6bb3be2e5c9381a77ef712a426353.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.8ex; height:3.509ex;" alt="{\textstyle \rho ={\sqrt {a^{2}+b^{2}}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\operatorname {atan2} (b,a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta =\operatorname {atan2} (b,a)}</annotation>

</semantics>

</math><img src="./f142b2384cd980378ee1468dac7995cc0e4d5b5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.944ex; height:2.843ex;" alt="{\displaystyle \theta =\operatorname {atan2} (b,a)}" loading="lazy">, where ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {atan2} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {atan2} }</annotation>

</semantics>

</math><img src="./474078eadf601ac09d580227d85df6f4e16e747f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.685ex; height:2.176ex;" alt="{\displaystyle \operatorname {atan2} }" loading="lazy">⁠ is the <a href="Atan2" title="Atan2">two-argument arctangent</a> function.</li>

<li><a href="Complex_logarithm" title="Complex logarithm">Logarithm</a> of z. The <a href="Principal_value" title="Principal value">principal value</a> of this logarithm is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log z=\ln \rho +i\theta ,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log z=\ln \rho +i\theta ,}</annotation>

</semantics>

</math><img src="./0fa84cbb8ff0ca04bbf34c505894bb17ac197a1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.454ex; height:2.676ex;" alt="{\displaystyle \log z=\ln \rho +i\theta ,}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ln }</annotation>

</semantics>

</math><img src="./c0de5ba4f372ede555d00035e70c50ed0b9625d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \ln }" loading="lazy"> denotes the <a href="Natural_logarithm" title="Natural logarithm">natural logarithm</a>. The other values of the logarithm are obtained by adding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2ik\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2ik\pi }</annotation>

</semantics>

</math><img src="./cf8c7b5a220c083f4b940a5932d4d73946517935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.508ex; height:2.176ex;" alt="{\displaystyle 2ik\pi }" loading="lazy"> for any integer k.</li>

<li>Canonical form of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w\log z.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w\log z.}</annotation>

</semantics>

</math><img src="./76ea2fc0f66a7fafb6a9c1c26f90b300668f79c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.145ex; height:2.509ex;" alt="{\displaystyle w\log z.}" loading="lazy"> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=c+di}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w=c+di}</annotation>

</semantics>

</math><img src="./af34dcf8f54e7235919f656300f9bde11e96a873.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.628ex; height:2.343ex;" alt="{\displaystyle w=c+di}" loading="lazy"> with c and d real, the values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w\log z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w\log z}</annotation>

</semantics>

</math><img src="./b1f8d9a8e88e0f2eaaf53ddb8b3d16442f5443bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.498ex; height:2.509ex;" alt="{\displaystyle w\log z}" loading="lazy"> are <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w\log z=(c\ln \rho -d\theta -2dk\pi )+i(d\ln \rho +c\theta +2ck\pi ),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w\log z=(c\ln \rho -d\theta -2dk\pi )+i(d\ln \rho +c\theta +2ck\pi ),}</annotation>

</semantics>

</math> the principal value corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=0.}</annotation>

</semantics>

</math><img src="./274c79b3ac027e0d0cd04cf86ae43c15567ba0bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.119ex; height:2.176ex;" alt="{\displaystyle k=0.}" loading="lazy"></li>

<li>Final result. Using the identities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x+y}=e^{x}e^{y}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{x+y}=e^{x}e^{y}}</annotation>

</semantics>

</math><img src="./7faf235fba706d2e2a1050fbca51de12294405b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.839ex; height:2.509ex;" alt="{\displaystyle e^{x+y}=e^{x}e^{y}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{y\ln x}=x^{y},}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{y\ln x}=x^{y},}</annotation>

</semantics>

</math><img src="./8cf1e02e72ba614178a80e52a1970c012eff32ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.343ex; height:3.009ex;" alt="{\displaystyle e^{y\ln x}=x^{y},}" loading="lazy"> one gets <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{w}=\rho ^{c}e^{-d(\theta +2k\pi )}\left(\cos(d\ln \rho +c\theta +2ck\pi )+i\sin(d\ln \rho +c\theta +2ck\pi )\right),}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{w}=\rho ^{c}e^{-d(\theta +2k\pi )}\left(\cos(d\ln \rho +c\theta +2ck\pi )+i\sin(d\ln \rho +c\theta +2ck\pi )\right),}</annotation>

</semantics>

</math> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=0}</annotation>

</semantics>

</math><img src="./6307c8a99dad7d0bcb712352ae0a748bd99a038b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}" loading="lazy"> for the principal value.</li></ul>

<div class="mw-heading mw-heading5"><h5 id="Examples">Examples</h5></div>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i^{i}}</annotation>

</semantics>

</math><img src="./7d4035ae18d005ce695d4fc30cb23c0771ab4cd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.602ex; height:2.676ex;" alt="{\displaystyle i^{i}}" loading="lazy"> The polar form of i is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=e^{i\pi /2},}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=e^{i\pi /2},}</annotation>

</semantics>

</math><img src="./c72ac8c3f1fa8a070f665f28405a41b7a5ca3c8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.017ex; height:3.176ex;" alt="{\displaystyle i=e^{i\pi /2},}" loading="lazy"> and the values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log i}</annotation>

</semantics>

</math><img src="./c6819e82b0d36458a266e0fc2a1048f01026bc4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.161ex; height:2.509ex;" alt="{\displaystyle \log i}" loading="lazy"> are thus <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log i=i\left({\frac {\pi }{2}}+2k\pi \right).}">

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log i=i\left({\frac {\pi }{2}}+2k\pi \right).}</annotation>

</semantics>

</math> It follows that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{i}=e^{i\log i}=e^{-{\frac {\pi }{2}}}e^{-2k\pi }.}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i^{i}=e^{i\log i}=e^{-{\frac {\pi }{2}}}e^{-2k\pi }.}</annotation>

</semantics>

</math>So, all values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{i}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i^{i}}</annotation>

</semantics>

</math><img src="./7d4035ae18d005ce695d4fc30cb23c0771ab4cd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.602ex; height:2.676ex;" alt="{\displaystyle i^{i}}" loading="lazy"> are real, the principal one being <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-{\frac {\pi }{2}}}\approx 0.2079.}">

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{-{\frac {\pi }{2}}}\approx 0.2079.}</annotation>

</semantics>

</math></li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-2)^{3+4i}}</annotation>

</semantics>

</math><img src="./8eb5be71bc68c8a9e30ae26fea962d1cae009b08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.502ex; height:3.176ex;" alt="{\displaystyle (-2)^{3+4i}}" loading="lazy"> Similarly, the polar form of −2 is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2=2e^{i\pi }.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -2=2e^{i\pi }.}</annotation>

</semantics>

</math><img src="./1b2bd6c0a239c9710293292058d5fedebdb13c04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.703ex; height:2.843ex;" alt="{\displaystyle -2=2e^{i\pi }.}" loading="lazy"> So, the above described method gives the values <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(-2)^{3+4i}&=2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2+3(\pi +2k\pi ))+i\sin(4\ln 2+3(\pi +2k\pi )))\\&=-2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2)+i\sin(4\ln 2)).\end{aligned}}}">

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(-2)^{3+4i}&=2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2+3(\pi +2k\pi ))+i\sin(4\ln 2+3(\pi +2k\pi )))\\&=-2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2)+i\sin(4\ln 2)).\end{aligned}}}</annotation>

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</math>In this case, all the values have the same argument <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\ln 2,}">

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<annotation encoding="application/x-tex">{\displaystyle 4\ln 2,}</annotation>

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</math><img src="./88a36542f2e27dce5330e0d8c0165972361fcf1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.685ex; height:2.509ex;" alt="{\displaystyle 4\ln 2,}" loading="lazy"> and different absolute values.</li></ul>

In both examples, all values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{w}}">

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<annotation encoding="application/x-tex">{\displaystyle z^{w}}</annotation>

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</math><img src="./3010585a40a0ee9155d34b74b5029fd163197dce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.343ex;" alt="{\displaystyle z^{w}}" loading="lazy"> have the same argument. More generally, this is true if and only if the <a href="Real_part" class="mw-redirect" title="Real part">real part</a> of w is an integer.

<div class="mw-heading mw-heading4"><h4 id="Failure_of_power_and_logarithm_identities">Failure of power and logarithm identities</h4></div>

Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined as single-valued functions. For example:

<div><ul><li>The identity log(bx) = x ⋅ log b holds whenever b is a positive real number and x is a real number. But for the <a href="Principal_branch" title="Principal branch">principal branch</a> of the complex logarithm one has

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<annotation encoding="application/x-tex">{\displaystyle \log((-i)^{2})=\log(-1)=i\pi \neq 2\log(-i)=2\log(e^{-i\pi /2})=2\,{\frac {-i\pi }{2}}=-i\pi }</annotation>

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Regardless of which branch of the logarithm is used, a similar failure of the identity will exist. The best that can be said (if only using this result) is that:

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<annotation encoding="application/x-tex">{\displaystyle \log w^{z}\equiv z\log w{\pmod {2\pi i}}}</annotation>

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This identity does not hold even when considering log as a multivalued function. The possible values of log(wz) contain those of z ⋅ log w as a <a href="Proper_subset" class="mw-redirect" title="Proper subset">proper subset</a>. Using Log(w) for the principal value of log(w) and m, n as any integers the possible values of both sides are:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left\{\log w^{z}\right\}&=\left\{z\cdot \operatorname {Log} w+z\cdot 2\pi in+2\pi im\mid m,n\in \mathbb {Z} \right\}\\\left\{z\log w\right\}&=\left\{z\operatorname {Log} w+z\cdot 2\pi in\mid n\in \mathbb {Z} \right\}\end{aligned}}}</annotation>

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</math></li><li>The identities (bc)x = bxcx and (b/c)x = bx/cx are valid when b and c are positive real numbers and x is a real number. But, for the principal values, one has

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<annotation encoding="application/x-tex">{\displaystyle (-1\cdot -1)^{\frac {1}{2}}=1\neq (-1)^{\frac {1}{2}}(-1)^{\frac {1}{2}}=i\cdot i=i^{2}=-1}</annotation>

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and

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {1}{-1}}\right)^{\frac {1}{2}}=(-1)^{\frac {1}{2}}=i\neq {\frac {1^{\frac {1}{2}}}{(-1)^{\frac {1}{2}}}}={\frac {1}{i}}=-i}</annotation>

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On the other hand, when x is an integer, the identities are valid for all nonzero complex numbers.

If exponentiation is considered as a multivalued function then the possible values of (−1 ⋅ −1)1/2 are {1, −1}. The identity holds, but saying {1} = {(−1 ⋅ −1)1/2} is incorrect.</li><li>The identity (ex)y = exy holds for real numbers x and y, but assuming its truth for complex numbers leads to the following <a href="Mathematical_fallacy" title="Mathematical fallacy">paradox</a>, discovered in 1827 by <a href="Thomas_Clausen_(mathematician)" title="Thomas Clausen (mathematician)">Clausen</a>:<a href="#cite_note-Clausen1827-38">[37]</a>

For any integer n, we have:

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<annotation encoding="application/x-tex">{\displaystyle e^{1+2\pi in}=e^{1}e^{2\pi in}=e\cdot 1=e}</annotation>

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</math><img src="./88b04065aeeb6eb592831d216a450d9f88dbfecd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:27.809ex; height:2.676ex;" alt="{\displaystyle e^{1+2\pi in}=e^{1}e^{2\pi in}=e\cdot 1=e}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle \left(e^{1+2\pi in}\right)^{1+2\pi in}=e\qquad }</annotation>

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</math><img src="./6e1954038cde6ecc51f28ea75ec0d7e66f497c77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.341ex; height:3.843ex;" alt="{\displaystyle \left(e^{1+2\pi in}\right)^{1+2\pi in}=e\qquad }" loading="lazy"> (taking the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+2\pi in)}">

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<annotation encoding="application/x-tex">{\displaystyle (1+2\pi in)}</annotation>

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</math><img src="./dc91367e144d6d812e07b85c34bc38099440a4be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.504ex; height:2.843ex;" alt="{\displaystyle (1+2\pi in)}" loading="lazy">-th power of both sides)</li>

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<annotation encoding="application/x-tex">{\displaystyle e^{1+4\pi in-4\pi ^{2}n^{2}}=e\qquad }</annotation>

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</math><img src="./c3abe61ab6846a0d8c21151865e509e146bf919c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.254ex; height:3.009ex;" alt="{\displaystyle e^{1+4\pi in-4\pi ^{2}n^{2}}=e\qquad }" loading="lazy"> (using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(e^{x}\right)^{y}=e^{xy}}">

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<annotation encoding="application/x-tex">{\displaystyle \left(e^{x}\right)^{y}=e^{xy}}</annotation>

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</math><img src="./dc118df35c28a26eddece8200757cd9e7f6fb7b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.286ex; height:3.009ex;" alt="{\displaystyle \left(e^{x}\right)^{y}=e^{xy}}" loading="lazy"> and expanding the exponent)</li>

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<annotation encoding="application/x-tex">{\displaystyle e^{1}e^{4\pi in}e^{-4\pi ^{2}n^{2}}=e\qquad }</annotation>

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</math><img src="./26c01def215de50bb5bee40a72cba2ef69754408.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.607ex; height:3.009ex;" alt="{\displaystyle e^{1}e^{4\pi in}e^{-4\pi ^{2}n^{2}}=e\qquad }" loading="lazy"> (using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x+y}=e^{x}e^{y}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{x+y}=e^{x}e^{y}}</annotation>

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</math><img src="./7faf235fba706d2e2a1050fbca51de12294405b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.839ex; height:2.509ex;" alt="{\displaystyle e^{x+y}=e^{x}e^{y}}" loading="lazy">)</li>

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<annotation encoding="application/x-tex">{\displaystyle e^{-4\pi ^{2}n^{2}}=1\qquad }</annotation>

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</math><img src="./e0759e20afd8c98073a934c1aa510c4f42ff63cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.915ex; height:3.009ex;" alt="{\displaystyle e^{-4\pi ^{2}n^{2}}=1\qquad }" loading="lazy"> (dividing by e)</li></ol>

but this is false when the integer n is nonzero.

The error is the following: by definition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{y}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{y}}</annotation>

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</math><img src="./7ff26085237c8dc6802eba0882a8aea22e890183.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.133ex; height:2.343ex;" alt="{\displaystyle e^{y}}" loading="lazy"> is a notation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(y),}">

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<annotation encoding="application/x-tex">{\displaystyle \exp(y),}</annotation>

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</math><img src="./6e1936b42c993855b3ce462c7b68e994fbedcd54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.164ex; height:2.843ex;" alt="{\displaystyle \exp(y),}" loading="lazy"> a true function, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{y}}">

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</math><img src="./c8561c712e86598255e8434a70affa18ffd7e0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.379ex; height:2.343ex;" alt="{\displaystyle x^{y}}" loading="lazy"> is a notation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(y\log x),}">

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<annotation encoding="application/x-tex">{\displaystyle \exp(y\log x),}</annotation>

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</math><img src="./490dfab20a13c16c3291f2ef5f4b2a00d946f52d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.24ex; height:2.843ex;" alt="{\displaystyle \exp(y\log x),}" loading="lazy"> which is a multi-valued function. Thus the notation is ambiguous when x = e. Here, before expanding the exponent, the second line should be

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<annotation encoding="application/x-tex">{\displaystyle \exp \left((1+2\pi in)\log \exp(1+2\pi in)\right)=\exp(1+2\pi in).}</annotation>

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Therefore, when expanding the exponent, one has implicitly supposed that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log \exp z=z}">

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<annotation encoding="application/x-tex">{\displaystyle \log \exp z=z}</annotation>

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</math><img src="./0ebe41c26ece7289dc4644f76f332c66ac318894.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.573ex; height:2.509ex;" alt="{\displaystyle \log \exp z=z}" loading="lazy"> for complex values of z, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity (ex)y = exy must be replaced by the identity

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which is a true identity between multivalued functions.</li></ul></div>

<div class="mw-heading mw-heading2"><h2 id="Irrationality_and_transcendence">Irrationality and transcendence</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Gelfond%E2%80%93Schneider_theorem" title="Gelfond–Schneider theorem">Gelfond–Schneider theorem</a></div>

If b is a positive real <a href="Algebraic_number" title="Algebraic number">algebraic number</a>, and x is a rational number, then bx is an algebraic number. This results from the theory of <a href="Algebraic_extension" title="Algebraic extension">algebraic extensions</a>. This remains true if b is any algebraic number, in which case, all values of bx (as a <a href="Multivalued_function" title="Multivalued function">multivalued function</a>) are algebraic. If x is <a href="Irrational_number" title="Irrational number">irrational</a> (that is, not rational), and both b and x are algebraic, Gelfond–Schneider theorem asserts that all values of bx are <a href="Transcendental_number" title="Transcendental number">transcendental</a> (that is, not algebraic), except if b equals 0 or 1.

In other words, if x is irrational and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\not \in \{0,1\},}">

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</math><img src="./146d85efbc22f5f38c9bc8556198014ed26494dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.169ex; height:2.843ex;" alt="{\displaystyle b\not \in \{0,1\},}" loading="lazy"> then at least one of b, x and bx is transcendental.

<div class="mw-heading mw-heading2"><h2 id="Integer_powers_in_algebra">Integer powers in algebra</h2></div>

The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any <a href="Associative_operation" class="mw-redirect" title="Associative operation">associative operation</a> denoted as a multiplication.<a href="#cite_note-39">[nb 2]</a> The definition of x0 requires further the existence of a <a href="Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">multiplicative identity</a>.<a href="#cite_note-40">[38]</a>

An <a href="Algebraic_structure" title="Algebraic structure">algebraic structure</a> consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by 1 is a <a href="Monoid" title="Monoid">monoid</a>. In such a monoid, exponentiation of an element x is defined inductively by

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<annotation encoding="application/x-tex">{\displaystyle x^{n+1}=xx^{n}}</annotation>

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</math><img src="./837c9728ee206eb7b79a82459266344695432df7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.625ex; height:2.676ex;" alt="{\displaystyle x^{n+1}=xx^{n}}" loading="lazy"> for every nonnegative integer n.</li></ul>

If n is a negative integer, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}}">

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</math><img src="./150d38e238991bc4d0689ffc9d2a852547d2658d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.548ex; height:2.343ex;" alt="{\displaystyle x^{n}}" loading="lazy"> is defined only if x has a <a href="Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>.<a href="#cite_note-41">[39]</a> In this case, the inverse of x is denoted x−1, and xn is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x^{-1}\right)^{-n}.}">

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Exponentiation with integer exponents obeys the following laws, for x and y in the algebraic structure, and m and n integers:

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<mtext>and, in particular, if the multiplication is commutative.</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x^{0}&=1\\x^{m+n}&=x^{m}x^{n}\\(x^{m})^{n}&=x^{mn}\\(xy)^{n}&=x^{n}y^{n}\quad {\text{if }}xy=yx,{\text{and, in particular, if the multiplication is commutative.}}\end{aligned}}}</annotation>

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</math><img src="./575c4faf2d2345f7d329b2ff309981fd92e79359.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:83.433ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}x^{0}&=1\\x^{m+n}&=x^{m}x^{n}\\(x^{m})^{n}&=x^{mn}\\(xy)^{n}&=x^{n}y^{n}\quad {\text{if }}xy=yx,{\text{and, in particular, if the multiplication is commutative.}}\end{aligned}}}" loading="lazy"></dd></dl>

These definitions are widely used in many areas of mathematics, notably for <a href="Group_(mathematics)" title="Group (mathematics)">groups</a>, <a href="Ring_(mathematics)" title="Ring (mathematics)">rings</a>, <a href="Field_(mathematics)" title="Field (mathematics)">fields</a>, <a href="Square_matrix" title="Square matrix">square matrices</a> (which form a ring). They apply also to <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> from a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> to itself, which form a monoid under <a href="Function_composition" title="Function composition">function composition</a>. This includes, as specific instances, <a href="Geometric_transformation" title="Geometric transformation">geometric transformations</a>, and <a href="Endomorphism" title="Endomorphism">endomorphisms</a> of any <a href="Mathematical_structure" title="Mathematical structure">mathematical structure</a>.

When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, if f is a <a href="Real_function" class="mw-redirect" title="Real function">real function</a> whose valued can be multiplied, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle f^{n}}</annotation>

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</math><img src="./2af7f3f2e54e4e35e26230a4e131c45e5d4000c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.539ex; height:2.676ex;" alt="{\displaystyle f^{n}}" loading="lazy"> denotes the exponentiation with respect of multiplication, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{\circ n}}">

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<annotation encoding="application/x-tex">{\displaystyle f^{\circ n}}</annotation>

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</math><img src="./04ab747aca28bc7831126a96302421c73a9acc8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.361ex; height:2.676ex;" alt="{\displaystyle f^{\circ n}}" loading="lazy"> may denote exponentiation with respect of <a href="Function_composition" title="Function composition">function composition</a>. That is,

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<annotation encoding="application/x-tex">{\displaystyle (f^{n})(x)=(f(x))^{n}=f(x)\,f(x)\cdots f(x),}</annotation>

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</math><img src="./292c822db243bb1b37262c71ea32f6e074d2060f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.914ex; height:2.843ex;" alt="{\displaystyle (f^{n})(x)=(f(x))^{n}=f(x)\,f(x)\cdots f(x),}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle (f^{\circ n})(x)=f(f(\cdots f(f(x))\cdots )).}</annotation>

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</math><img src="./3befa02e0ff1e5213ebe9d9b6913e5a9e371fdc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.956ex; height:2.843ex;" alt="{\displaystyle (f^{\circ n})(x)=f(f(\cdots f(f(x))\cdots )).}" loading="lazy"></dd></dl>

Commonly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f^{n})(x)}">

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<annotation encoding="application/x-tex">{\displaystyle (f^{n})(x)}</annotation>

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</math><img src="./66e5572d062599e0bd084eb121360338cdc425ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.487ex; height:2.843ex;" alt="{\displaystyle (f^{n})(x)}" loading="lazy"> is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)^{n},}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(x)^{n},}</annotation>

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</math><img src="./b959cdee7280f6e694298055536de96b47f846bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.283ex; height:2.843ex;" alt="{\displaystyle f(x)^{n},}" loading="lazy"> while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f^{\circ n})(x)}">

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (f^{\circ n})(x)}</annotation>

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</math><img src="./e6919453ef0e2eb0aaf70492b1bb9d55ab2f123d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.309ex; height:2.843ex;" alt="{\displaystyle (f^{\circ n})(x)}" loading="lazy"> is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{n}(x).}">

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<annotation encoding="application/x-tex">{\displaystyle f^{n}(x).}</annotation>

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</math><img src="./b6b5becb1d2856a002c16a04188b83188fe092ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.325ex; height:2.843ex;" alt="{\displaystyle f^{n}(x).}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="In_a_group">In a group</h3></div>

A <a href="Multiplicative_group" title="Multiplicative group">multiplicative group</a> is a set with as <a href="Associative_operation" class="mw-redirect" title="Associative operation">associative operation</a> denoted as multiplication, that has an <a href="Identity_element" title="Identity element">identity element</a>, and such that every element has an inverse.

So, if G is a group, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle x^{n}}</annotation>

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</math><img src="./150d38e238991bc4d0689ffc9d2a852547d2658d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.548ex; height:2.343ex;" alt="{\displaystyle x^{n}}" loading="lazy"> is defined for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in G}">

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<annotation encoding="application/x-tex">{\displaystyle x\in G}</annotation>

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</math><img src="./6d7e0b51bd905f35d11790939139d18014f8b017.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.997ex; height:2.176ex;" alt="{\displaystyle x\in G}" loading="lazy"> and every integer n.

The set of all powers of an element of a group form a <a href="Subgroup" title="Subgroup">subgroup</a>. A group (or subgroup) that consists of all powers of a specific element x is the <a href="Cyclic_group" title="Cyclic group">cyclic group</a> generated by x. If all the powers of x are distinct, the group is <a href="Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to the <a href="Additive_group" title="Additive group">additive group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

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</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"> of the integers. Otherwise, the cyclic group is <a href="Finite_group" title="Finite group">finite</a> (it has a finite number of elements), and its number of elements is the <a href="Order_(group_theory)" title="Order (group theory)">order</a> of x. If the order of x is n, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}=x^{0}=1,}">

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<annotation encoding="application/x-tex">{\displaystyle x^{n}=x^{0}=1,}</annotation>

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</math><img src="./3cc097b59de85cade1d80c1505c9128135b95181.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.938ex; height:3.009ex;" alt="{\displaystyle x^{n}=x^{0}=1,}" loading="lazy"> and the cyclic group generated by x consists of the n first powers of x (starting indifferently from the exponent 0 or 1).

Order of elements play a fundamental role in <a href="Group_theory" title="Group theory">group theory</a>. For example, the order of an element in a finite group is always a divisor of the number of elements of the group (the order of the group). The possible orders of group elements are important in the study of the structure of a group (see <a href="Sylow_theorems" title="Sylow theorems">Sylow theorems</a>), and in the <a href="Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a>.

Superscript notation is also used for <a href="Conjugacy_class" title="Conjugacy class">conjugation</a>; that is, gh = h−1gh, where g and h are elements of a group. This notation cannot be confused with exponentiation, since the superscript is not an integer. The motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g^{h})^{k}=g^{hk}}">

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<annotation encoding="application/x-tex">{\displaystyle (g^{h})^{k}=g^{hk}}</annotation>

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</math><img src="./f1f460a347be8231f0caf04ce5ca180c0ef47c38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.447ex; height:3.176ex;" alt="{\displaystyle (g^{h})^{k}=g^{hk}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (gh)^{k}=g^{k}h^{k}.}">

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<annotation encoding="application/x-tex">{\displaystyle (gh)^{k}=g^{k}h^{k}.}</annotation>

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</math><img src="./1972d6c769000910c35de759c37a803404407abb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.733ex; height:3.176ex;" alt="{\displaystyle (gh)^{k}=g^{k}h^{k}.}" loading="lazy">

In a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>, it may occur that some nonzero elements satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}=0}">

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<annotation encoding="application/x-tex">{\displaystyle x^{n}=0}</annotation>

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</math><img src="./05c1b8ff8fb258b6d0285a828d58ee8eaf4aaf07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.809ex; height:2.343ex;" alt="{\displaystyle x^{n}=0}" loading="lazy"> for some integer n. Such an element is said to be <a href="Nilpotent" title="Nilpotent">nilpotent</a>. In a <a href="Commutative_ring" title="Commutative ring">commutative ring</a>, the nilpotent elements form an <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a>, called the <a href="Nilradical_of_a_ring" title="Nilradical of a ring">nilradical</a> of the ring.

If the nilradical is reduced to the <a href="Zero_ideal" class="mw-redirect" title="Zero ideal">zero ideal</a> (that is, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\neq 0}</annotation>

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</math><img src="./35a455db7b2aab1b0e72ccbc7385e4424e2372e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\displaystyle x\neq 0}" loading="lazy"> implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle x^{n}\neq 0}</annotation>

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</math><img src="./b9131d763fff10f0a8dacf6cc14d5c8ded069b2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.809ex; height:2.843ex;" alt="{\displaystyle x^{n}\neq 0}" loading="lazy"> for every positive integer n), the commutative ring is said to be <a href="Reduced_ring" title="Reduced ring">reduced</a>. Reduced rings are important in <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, since the <a href="Coordinate_ring" class="mw-redirect" title="Coordinate ring">coordinate ring</a> of an <a href="Affine_algebraic_set" class="mw-redirect" title="Affine algebraic set">affine algebraic set</a> is always a reduced ring.

More generally, given an ideal I in a commutative ring R, the set of the elements of R that have a power in I is an ideal, called the <a href="Radical_of_an_ideal" title="Radical of an ideal">radical</a> of I. The nilradical is the radical of the <a href="Zero_ideal" class="mw-redirect" title="Zero ideal">zero ideal</a>. A <a href="Radical_ideal" class="mw-redirect" title="Radical ideal">radical ideal</a> is an ideal that equals its own radical. In a <a href="Polynomial_ring" title="Polynomial ring">polynomial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[x_{1},\ldots ,x_{n}]}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle k[x_{1},\ldots ,x_{n}]}</annotation>

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</math><img src="./37c2b680cd4b215ac5c3c548a0e596d534526cab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.615ex; height:2.843ex;" alt="{\displaystyle k[x_{1},\ldots ,x_{n}]}" loading="lazy"> over a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> k, an ideal is radical if and only if it is the set of all polynomials that are zero on an affine algebraic set (this is a consequence of <a href="Hilbert's_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a>).

<div class="mw-heading mw-heading3"><h3 id="Matrices_and_linear_operators">Matrices and linear operators</h3></div>

If A is a square matrix, then the product of A with itself n times is called the <a href="Matrix_power" class="mw-redirect" title="Matrix power">matrix power</a>. Also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{0}}">

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<annotation encoding="application/x-tex">{\displaystyle A^{0}}</annotation>

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</math><img src="./46d16b5c2b54b866287fd4cbc26a180ffb93978a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.676ex;" alt="{\displaystyle A^{0}}" loading="lazy"> is defined to be the identity matrix,<a href="#cite_note-42">[40]</a> and if A is invertible, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{-n}=\left(A^{-1}\right)^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle A^{-n}=\left(A^{-1}\right)^{n}}</annotation>

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</math><img src="./e7d3b01ba9ec1ac82badfdc7b3162aea150897cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.763ex; height:3.343ex;" alt="{\displaystyle A^{-n}=\left(A^{-1}\right)^{n}}" loading="lazy">.

Matrix powers appear often in the context of <a href="Discrete_dynamical_system" class="mw-redirect" title="Discrete dynamical system">discrete dynamical systems</a>, where the matrix A expresses a transition from a state vector x of some system to the next state Ax of the system.<a href="#cite_note-43">[41]</a> This is the standard interpretation of a <a href="Markov_chain" title="Markov chain">Markov chain</a>, for example. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{2}x}">

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<annotation encoding="application/x-tex">{\displaystyle A^{2}x}</annotation>

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</math><img src="./9d33e9736b0631f1140485c31f200097d608310c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.127ex; height:2.676ex;" alt="{\displaystyle A^{2}x}" loading="lazy"> is the state of the system after two time steps, and so forth: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{n}x}">

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<annotation encoding="application/x-tex">{\displaystyle A^{n}x}</annotation>

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</math><img src="./3129f24e0438721a140ebb8463225980b9e1816e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.291ex; height:2.343ex;" alt="{\displaystyle A^{n}x}" loading="lazy"> is the state of the system after n time steps. The matrix power <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{n}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle A^{n}}</annotation>

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</math><img src="./f40057ac796fcce575670828684300e42bf8a227.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.962ex; height:2.343ex;" alt="{\displaystyle A^{n}}" loading="lazy"> is the transition matrix between the state now and the state at a time n steps in the future. So computing matrix powers is equivalent to solving the evolution of the dynamical system. In many cases, matrix powers can be expediently computed by using <a href="Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues and eigenvectors</a>.

Apart from matrices, more general <a href="Linear_operator" class="mw-redirect" title="Linear operator">linear operators</a> can also be exponentiated. An example is the <a href="Derivative" title="Derivative">derivative</a> operator of calculus, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d/dx}">

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<annotation encoding="application/x-tex">{\displaystyle d/dx}</annotation>

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</math><img src="./75a0e680edceb47b7d233535262fcacd931585f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.924ex; height:2.843ex;" alt="{\displaystyle d/dx}" loading="lazy">, which is a linear operator acting on functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

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</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> to give a new function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (d/dx)f(x)=f'(x)}">

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<annotation encoding="application/x-tex">{\displaystyle (d/dx)f(x)=f'(x)}</annotation>

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</math><img src="./aa085701eb7dddf0f579fc29875f4499f2076dbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.393ex; height:3.009ex;" alt="{\displaystyle (d/dx)f(x)=f'(x)}" loading="lazy">. The nth power of the differentiation operator is the nth derivative:

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {d}{dx}}\right)^{n}f(x)={\frac {d^{n}}{dx^{n}}}f(x)=f^{(n)}(x).}</annotation>

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</math><img src="./d054736ab441db10f48e6ffcc08f1999b3c68e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.258ex; height:6.176ex;" alt="{\displaystyle \left({\frac {d}{dx}}\right)^{n}f(x)={\frac {d^{n}}{dx^{n}}}f(x)=f^{(n)}(x).}" loading="lazy"></dd></dl>

These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory of <a href="C0-semigroup" title="C0-semigroup">semigroups</a>.<a href="#cite_note-44">[42]</a> Just as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics. Examples include approaches to solving the <a href="Heat_equation" title="Heat equation">heat equation</a>, <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>, <a href="Wave_equation" title="Wave equation">wave equation</a>, and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the <a href="Fractional_derivative" class="mw-redirect" title="Fractional derivative">fractional derivative</a> which, together with the <a href="Fractional_integral" class="mw-redirect" title="Fractional integral">fractional integral</a>, is one of the basic operations of the <a href="Fractional_calculus" title="Fractional calculus">fractional calculus</a>.

<div class="mw-heading mw-heading3"><h3 id="Finite_fields">Finite fields</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Finite_field" title="Finite field">Finite field</a></div>

A <a href="Field_(mathematics)" title="Field (mathematics)">field</a> is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication is <a href="Associative" class="mw-redirect" title="Associative">associative</a> and every nonzero element has a <a href="Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>. This implies that exponentiation with integer exponents is well-defined, except for nonpositive powers of 0. Common examples are the field of <a href="Complex_number" title="Complex number">complex numbers</a>, the <a href="Real_number" title="Real number">real numbers</a> and the <a href="Rational_number" title="Rational number">rational numbers</a>, considered earlier in this article, which are all <a href="Infinite_set" title="Infinite set">infinite</a>.

A finite field is a field with a <a href="Finite_set" title="Finite set">finite number</a> of elements. This number of elements is either a <a href="Prime_number" title="Prime number">prime number</a> or a <a href="Prime_power" title="Prime power">prime power</a>; that is, it has the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=p^{k},}">

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</math><img src="./d33e606a2140ec546aae29e5eaed57614811168a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.073ex; height:3.009ex;" alt="{\displaystyle q=p^{k},}" loading="lazy"> where p is a prime number, and k is a positive integer. For every such q, there are fields with q elements. The fields with q elements are all <a href="Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a>, which allows, in general, working as if there were only one field with q elements, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}.}</annotation>

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</math><img src="./3b617359f23f7534bdd247769e83a120fb155797.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.056ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}.}" loading="lazy">

One has

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</math><img src="./e6bc709662a3fbdf3cdb7001c63e8dd9f4dc19d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.746ex; height:2.343ex;" alt="{\displaystyle x^{q}=x}" loading="lazy"></dd></dl>

for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {F} _{q}.}">

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<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {F} _{q}.}</annotation>

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</math><img src="./20834e63283d53b54806b7ca6eaf7ed18f0c06da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.226ex; height:2.843ex;" alt="{\displaystyle x\in \mathbb {F} _{q}.}" loading="lazy">

A <a href="Primitive_element_(finite_field)" title="Primitive element (finite field)">primitive element</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}}</annotation>

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</math><img src="./dbb96e056c071d13fc7702013f9273e7f5cd88a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}" loading="lazy"> is an element g such that the set of the q − 1 first powers of g (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{g^{1}=g,g^{2},\ldots ,g^{p-1}=g^{0}=1\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{g^{1}=g,g^{2},\ldots ,g^{p-1}=g^{0}=1\}}</annotation>

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</math><img src="./89078f12ece5f65a8dcafcc33ee3a8e1f9b557b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.906ex; height:3.176ex;" alt="{\displaystyle \{g^{1}=g,g^{2},\ldots ,g^{p-1}=g^{0}=1\}}" loading="lazy">) equals the set of the nonzero elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}.}</annotation>

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</math><img src="./3b617359f23f7534bdd247769e83a120fb155797.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.056ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}.}" loading="lazy"> There are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (p-1)}">

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</math><img src="./f94195099530fc8f38a0002acb153d40572013da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.502ex; height:2.843ex;" alt="{\displaystyle \varphi (p-1)}" loading="lazy"> primitive elements in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q},}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q},}</annotation>

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</math><img src="./4e7d145216d6fd8a79e6a869ac9e5c54c6334be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.056ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q},}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

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</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> is <a href="Euler's_totient_function" title="Euler's totient function">Euler's totient function</a>.

In <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q},}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q},}</annotation>

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</math><img src="./4e7d145216d6fd8a79e6a869ac9e5c54c6334be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.056ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q},}" loading="lazy"> the <a href="Freshman's_dream" title="Freshman's dream">freshman's dream</a> identity

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<annotation encoding="application/x-tex">{\displaystyle (x+y)^{p}=x^{p}+y^{p}}</annotation>

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</math><img src="./ac9c35247ed004b6fe4b3b4ca807e0e558a8a94f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.741ex; height:2.843ex;" alt="{\displaystyle (x+y)^{p}=x^{p}+y^{p}}" loading="lazy"></dd></dl>

is true for the exponent p. As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{p}=x}">

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<annotation encoding="application/x-tex">{\displaystyle x^{p}=x}</annotation>

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</math><img src="./da460092643abfe81093662660377ef0a133b85a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.817ex; height:2.343ex;" alt="{\displaystyle x^{p}=x}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q},}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q},}</annotation>

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</math><img src="./4e7d145216d6fd8a79e6a869ac9e5c54c6334be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.056ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q},}" loading="lazy"> It follows that the map

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F\colon {}&\mathbb {F} _{q}\to \mathbb {F} _{q}\\&x\mapsto x^{p}\end{aligned}}}</annotation>

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</math><img src="./8d42cbf34d993a19e4ba5d5e2ea141b95a7094b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.958ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}F\colon {}&\mathbb {F} _{q}\to \mathbb {F} _{q}\\&x\mapsto x^{p}\end{aligned}}}" loading="lazy"></dd></dl>

is <a href="Linear_map" title="Linear map">linear</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q},}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q},}</annotation>

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</math><img src="./4e7d145216d6fd8a79e6a869ac9e5c54c6334be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.056ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q},}" loading="lazy"> and is a <a href="Field_automorphism" class="mw-redirect" title="Field automorphism">field automorphism</a>, called the <a href="Frobenius_automorphism" class="mw-redirect" title="Frobenius automorphism">Frobenius automorphism</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=p^{k},}">

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</math><img src="./d33e606a2140ec546aae29e5eaed57614811168a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.073ex; height:3.009ex;" alt="{\displaystyle q=p^{k},}" loading="lazy"> the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}}</annotation>

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</math><img src="./dbb96e056c071d13fc7702013f9273e7f5cd88a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}" loading="lazy"> has k automorphisms, which are the k first powers (under <a href="Function_composition" title="Function composition">composition</a>) of F. In other words, the <a href="Galois_group" title="Galois group">Galois group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q}}</annotation>

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</math><img src="./dbb96e056c071d13fc7702013f9273e7f5cd88a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.409ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q}}" loading="lazy"> is <a href="Cyclic_group" title="Cyclic group">cyclic</a> of order k, generated by the Frobenius automorphism.

The <a href="Diffie%E2%80%93Hellman_key_exchange" title="Diffie–Hellman key exchange">Diffie–Hellman key exchange</a> is an application of exponentiation in finite fields that is widely used for <a href="Secure_communication" title="Secure communication">secure communications</a>. It uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the <a href="Discrete_logarithm" title="Discrete logarithm">discrete logarithm</a>, is computationally expensive. More precisely, if g is a primitive element in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{q},}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{q},}</annotation>

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</math><img src="./4e7d145216d6fd8a79e6a869ac9e5c54c6334be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.056ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} _{q},}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{e}}">

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</math><img src="./3d191a112f8ba1bf40408fe1d5a22fe9430b1725.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.117ex; height:2.676ex;" alt="{\displaystyle g^{e}}" loading="lazy"> can be efficiently computed with <a href="Exponentiation_by_squaring" title="Exponentiation by squaring">exponentiation by squaring</a> for any e, even if q is large, while there is no known computationally practical algorithm that allows retrieving e from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{e}}">

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</math><img src="./3d191a112f8ba1bf40408fe1d5a22fe9430b1725.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.117ex; height:2.676ex;" alt="{\displaystyle g^{e}}" loading="lazy"> if q is sufficiently large.

<div class="mw-heading mw-heading2"><h2 id="Powers_of_sets">Powers of sets </h2></div>

The <a href="Cartesian_product" title="Cartesian product">Cartesian product</a> of two <a href="Set_(mathematics)" title="Set (mathematics)">sets</a> S and T is the set of the <a href="Ordered_pair" title="Ordered pair">ordered pairs</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}">

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</math><img src="./41cf50e4a314ca8e2c30964baa8d26e5be7a9386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in S}">

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</math><img src="./51186ba8afb2067573a9082d55dd383df1ea9214.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.67ex; height:2.176ex;" alt="{\displaystyle x\in S}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in T.}">

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</math><img src="./5771df9ffc159b26f70de0457161c3219ddbfae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.279ex; height:2.509ex;" alt="{\displaystyle y\in T.}" loading="lazy"> This operation is not properly <a href="Commutative" class="mw-redirect" title="Commutative">commutative</a> nor <a href="Associative" class="mw-redirect" title="Associative">associative</a>, but has these properties <a href="Up_to" title="Up to">up to</a> <a href="Canonical_map" title="Canonical map">canonical</a> <a href="Isomorphism" title="Isomorphism">isomorphisms</a>, that allow identifying, for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,(y,z)),}">

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</math><img src="./d73989a253d715860d77fa635ae87ef49e8a35e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.907ex; height:2.843ex;" alt="{\displaystyle (x,(y,z)),}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((x,y),z),}">

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</math><img src="./a8069919ef5efb3f6a066fdd78dda061a2f80250.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.907ex; height:2.843ex;" alt="{\displaystyle ((x,y),z),}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z).}">

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</math><img src="./ae95f948320ab14e083d5645a2faab53eaf508ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.097ex; height:2.843ex;" alt="{\displaystyle (x,y,z).}" loading="lazy">

This allows defining the nth power <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}}">

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</math><img src="./ee006452a59bf1eb29983b4412348b66517a2d23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.74ex; height:2.343ex;" alt="{\displaystyle S^{n}}" loading="lazy"> of a set S as the set of all n-<a href="Tuple" title="Tuple">tuples</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\ldots ,x_{n})}">

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When S is endowed with some structure, it is frequent that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle S^{n}}</annotation>

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</math><img src="./ee006452a59bf1eb29983b4412348b66517a2d23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.74ex; height:2.343ex;" alt="{\displaystyle S^{n}}" loading="lazy"> is naturally endowed with a similar structure. In this case, the term "<a href="Direct_product" title="Direct product">direct product</a>" is generally used instead of "Cartesian product", and exponentiation denotes product structure. For example <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> denotes the real numbers) denotes the Cartesian product of n copies of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ,}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ,}</annotation>

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</math><img src="./0522388d36b55de7babe4bbfc49475eaf590c2bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} ,}" loading="lazy"> as well as their direct product as <a href="Vector_space" title="Vector space">vector space</a>, <a href="Topological_space" title="Topological space">topological spaces</a>, <a href="Ring_(mathematics)" title="Ring (mathematics)">rings</a>, etc.

<div class="mw-heading mw-heading3"><h3 id="Sets_as_exponents">Sets as exponents</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Function_(mathematics)#Set_exponentiation" title="Function (mathematics)">Function (mathematics) § Set exponentiation</a></div>

A n-tuple <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\ldots ,x_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{1},\ldots ,x_{n})}</annotation>

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</math><img src="./7935f7983d8a5ae59fea84efe65415235fa7c47b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{1},\ldots ,x_{n})}" loading="lazy"> of elements of S can be considered as a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,\ldots ,n\}.}">

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<annotation encoding="application/x-tex">{\displaystyle \{1,\ldots ,n\}.}</annotation>

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</math><img src="./c0b292d5de5f592fa4145ff633eebbbfbb6163a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.707ex; height:2.843ex;" alt="{\displaystyle \{1,\ldots ,n\}.}" loading="lazy"> This generalizes to the following notation.

Given two sets S and T, the set of all functions from T to S is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{T}}">

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</math><img src="./7c96ff2287e146f1ccab6ae915d9bba2643f7fb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.911ex; height:2.676ex;" alt="{\displaystyle S^{T}}" loading="lazy">. This exponential notation is justified by the following canonical isomorphisms (for the first one, see <a href="Currying" title="Currying">Currying</a>):

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<annotation encoding="application/x-tex">{\displaystyle (S^{T})^{U}\cong S^{T\times U},}</annotation>

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</math><img src="./f277973ecd060c67a5d360d9ed2709000fb90d41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.408ex; height:3.176ex;" alt="{\displaystyle (S^{T})^{U}\cong S^{T\times U},}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle S^{T\sqcup U}\cong S^{T}\times S^{U},}</annotation>

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</math><img src="./5fcfcf99530912fab6745aa560027d049c802cd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.779ex; height:3.009ex;" alt="{\displaystyle S^{T\sqcup U}\cong S^{T}\times S^{U},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }">

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<annotation encoding="application/x-tex">{\displaystyle \times }</annotation>

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</math><img src="./0ffafff1ad26cbe49045f19a67ce532116a32703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }" loading="lazy"> denotes the Cartesian product, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sqcup }">

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</math><img src="./1596aedf354da694149e44ce2bf53ede54eca8cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \sqcup }" loading="lazy"> the <a href="Disjoint_union" title="Disjoint union">disjoint union</a>.

One can use sets as exponents for other operations on sets, typically for <a href="Direct_sum" title="Direct sum">direct sums</a> of <a href="Abelian_group" title="Abelian group">abelian groups</a>, <a href="Vector_space" title="Vector space">vector spaces</a>, or <a href="Module_(mathematics)" title="Module (mathematics)">modules</a>. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{\mathbb {N} }}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{\mathbb {N} }}</annotation>

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</math><img src="./2608c429068c69675f1144f0ba8249603a8ceaf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.097ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{\mathbb {N} }}" loading="lazy"> denotes the vector space of the <a href="Infinite_sequence" class="mw-redirect" title="Infinite sequence">infinite sequences</a> of real numbers, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{(\mathbb {N} )}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{(\mathbb {N} )}}</annotation>

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</math><img src="./8efa88e23ba53b9c6994661a52e4f64de7484ad4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.376ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} ^{(\mathbb {N} )}}" loading="lazy"> the vector space of those sequences that have a finite number of nonzero elements. The latter has a <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> consisting of the sequences with exactly one nonzero element that equals 1, while the <a href="Hamel_basis" class="mw-redirect" title="Hamel basis">Hamel bases</a> of the former cannot be explicitly described (because their existence involves <a href="Zorn's_lemma" title="Zorn's lemma">Zorn's lemma</a>).

In this context, 2 can represents the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1\}.}">

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<annotation encoding="application/x-tex">{\displaystyle \{0,1\}.}</annotation>

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</math><img src="./7c11174aaa7a4b001c41084cd9d3abc7cbdf9d6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.331ex; height:2.843ex;" alt="{\displaystyle \{0,1\}.}" loading="lazy"> So, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{S}}">

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<annotation encoding="application/x-tex">{\displaystyle 2^{S}}</annotation>

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</math><img src="./0468338b3789428c4529defb8eb4a1faa4e9876b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.455ex; height:2.676ex;" alt="{\displaystyle 2^{S}}" loading="lazy"> denotes the <a href="Power_set" title="Power set">power set</a> of S, that is the set of the functions from S to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1\},}">

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<annotation encoding="application/x-tex">{\displaystyle \{0,1\},}</annotation>

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</math><img src="./af00fa53daf4f65aab7baf5ca959958b3b5853c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.331ex; height:2.843ex;" alt="{\displaystyle \{0,1\},}" loading="lazy"> which can be identified with the set of the <a href="Subset" title="Subset">subsets</a> of S, by mapping each function to the <a href="Inverse_image" class="mw-redirect" title="Inverse image">inverse image</a> of 1.

This fits in with the <a href="Cardinal_exponentiation" class="mw-redirect" title="Cardinal exponentiation">exponentiation of cardinal numbers</a>, in the sense that |ST| = |S||T|, where |X| is the cardinality of X.

<div class="mw-heading mw-heading3"><h3 id="In_category_theory">In category theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Cartesian_closed_category" title="Cartesian closed category">Cartesian closed category</a></div>

In the <a href="Category_of_sets" title="Category of sets">category of sets</a>, the <a href="Morphism" title="Morphism">morphisms</a> between sets X and Y are the functions from X to Y. It results that the set of the functions from X to Y that is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y^{X}}">

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</math><img src="./233b4c76e7ea7aad9de5488491e1c6c7363c0bea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.533ex; height:2.509ex;" alt="{\displaystyle Y^{X}}" loading="lazy"> in the preceding section can also be denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hom(X,Y).}">

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<annotation encoding="application/x-tex">{\displaystyle \hom(X,Y).}</annotation>

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</math><img src="./f3fd2200fce821a8e094df13538471c54b4d1c7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.634ex; height:2.843ex;" alt="{\displaystyle \hom(X,Y).}" loading="lazy"> The isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S^{T})^{U}\cong S^{T\times U}}">

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<annotation encoding="application/x-tex">{\displaystyle (S^{T})^{U}\cong S^{T\times U}}</annotation>

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</math><img src="./a313ea92062439daab14a07a711d284ea7ace14b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.761ex; height:3.176ex;" alt="{\displaystyle (S^{T})^{U}\cong S^{T\times U}}" loading="lazy"> can be rewritten

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<annotation encoding="application/x-tex">{\displaystyle \hom(U,S^{T})\cong \hom(T\times U,S).}</annotation>

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</math><img src="./9a5da53d0a34d7b09df9be4519ce53f0c1e75fbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.665ex; height:3.176ex;" alt="{\displaystyle \hom(U,S^{T})\cong \hom(T\times U,S).}" loading="lazy"></dd></dl>

This means the functor "exponentiation to the power T " is a <a href="Right_adjoint" class="mw-redirect" title="Right adjoint">right adjoint</a> to the functor "direct product with T ".

This generalizes to the definition of <a href="Exponential_(category_theory)" class="mw-redirect" title="Exponential (category theory)">exponentiation in a category</a> in which finite <a href="Direct_product" title="Direct product">direct products</a> exist: in such a category, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to X^{T}}">

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<annotation encoding="application/x-tex">{\displaystyle X\to X^{T}}</annotation>

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</math><img src="./e050b8864a3a80ec262a4462d502d934a019fc75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.98ex; height:2.676ex;" alt="{\displaystyle X\to X^{T}}" loading="lazy"> is, if it exists, a right adjoint to the functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\to T\times Y.}">

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<annotation encoding="application/x-tex">{\displaystyle Y\to T\times Y.}</annotation>

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</math><img src="./3cf1c1e43f974b88ce9ed8261cbdd0ead41d3084.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.284ex; height:2.176ex;" alt="{\displaystyle Y\to T\times Y.}" loading="lazy"> A category is called a Cartesian closed category, if direct products exist, and the functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\to X\times Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y\to X\times Y}</annotation>

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</math><img src="./4dc6188973439b11e5162bb6d37c3233d34fbcfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.981ex; height:2.176ex;" alt="{\displaystyle Y\to X\times Y}" loading="lazy"> has a right adjoint for every T.

<div class="mw-heading mw-heading2"><h2 id="Repeated_exponentiation">Repeated exponentiation</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Tetration" title="Tetration">Tetration</a> and <a href="Hyperoperation" title="Hyperoperation">Hyperoperation</a></div>

Just as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 or tetration. Iterating tetration leads to another operation, and so on, a concept named <a href="Hyperoperation" title="Hyperoperation">hyperoperation</a>. This sequence of operations is expressed by the <a href="Ackermann_function" title="Ackermann function">Ackermann function</a> and <a href="Knuth's_up-arrow_notation" title="Knuth's up-arrow notation">Knuth's up-arrow notation</a>. Just as exponentiation grows faster than multiplication, which is faster-growing than addition, tetration is faster-growing than exponentiation. Evaluated at (3, 3), the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and 7625597484987 (=327 = 333 = 33) respectively.

<div class="mw-heading mw-heading2"><h2 id="Limits_of_powers">Limits of powers</h2></div>

<a href="Zero_to_the_power_of_zero" title="Zero to the power of zero">Zero to the power of zero</a> gives a number of examples of limits that are of the <a href="Indeterminate_form" title="Indeterminate form">indeterminate form</a> 00. The limits in these examples exist, but have different values, showing that the two-variable function xy has no limit at the point (0, 0). One may consider at what points this function does have a limit.

More precisely, consider the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)=x^{y}}">

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</math><img src="./84423b7745fa7e979e05f4446981e058a2a84472.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.084ex; height:2.843ex;" alt="{\displaystyle f(x,y)=x^{y}}" loading="lazy"> defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=\{(x,y)\in \mathbf {R} ^{2}:x>0\}}">

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<annotation encoding="application/x-tex">{\displaystyle D=\{(x,y)\in \mathbf {R} ^{2}:x>0\}}</annotation>

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</math><img src="./2bb3ae1d04fc8a4e673d1694da043dfc7fc77193.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.102ex; height:3.176ex;" alt="{\displaystyle D=\{(x,y)\in \mathbf {R} ^{2}:x>0\}}" loading="lazy">. Then D can be viewed as a subset of R2 (that is, the set of all pairs (x, y) with x, y belonging to the <a href="Extended_real_number_line" title="Extended real number line">extended real number line</a> R = [−∞, +∞], endowed with the <a href="Product_topology" title="Product topology">product topology</a>), which will contain the points at which the function f has a limit.

In fact, f has a limit at all <a href="Accumulation_point" title="Accumulation point">accumulation points</a> of D, except for (0, 0), (+∞, 0), (1, +∞) and (1, −∞).<a href="#cite_note-45">[43]</a> Accordingly, this allows one to define the powers xy by continuity whenever 0 ≤ x ≤ +∞, −∞ ≤ y ≤ +∞, except for 00, (+∞)0, 1+∞ and 1−∞, which remain indeterminate forms.

Under this definition by continuity, we obtain:

These powers are obtained by taking limits of xy for positive values of x. This method does not permit a definition of xy when x < 0, since pairs (x, y) with x < 0 are not accumulation points of D.

On the other hand, when n is an integer, the power xn is already meaningful for all values of x, including negative ones. This may make the definition 0n = +∞ obtained above for negative n problematic when n is odd, since in this case xn → +∞ as x tends to 0 through positive values, but not negative ones.

<div class="mw-heading mw-heading2"><h2 id="Efficient_computation_with_integer_exponents">Efficient computation with integer exponents</h2></div>

Computing bn using iterated multiplication requires n − 1 multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2100, apply <a href="Horner's_rule" class="mw-redirect" title="Horner's rule">Horner's rule</a> to the exponent 100 written in binary:

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<annotation encoding="application/x-tex">{\displaystyle 100=2^{2}+2^{5}+2^{6}=2^{2}(1+2^{3}(1+2))}</annotation>

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</math><img src="./415ef98efcb0da84c62b10ec69ef2bd6de11e142.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.235ex; height:3.176ex;" alt="{\displaystyle 100=2^{2}+2^{5}+2^{6}=2^{2}(1+2^{3}(1+2))}" loading="lazy">.</dd></dl>

Then compute the following terms in order, reading Horner's rule from right to left.

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</style>

<td>22 = 4

</td></tr>

<tr>

<td>2 (22) = 23 = 8

</td></tr>

<tr>

<td>(23)2 = 26 = 64

</td></tr>

<tr>

<td>(26)2 = 212 = 4096

</td></tr>

<tr>

<td>(212)2 = 224 = 16777216

</td></tr>

<tr>

<td>2 (224) = 225 = 33554432

</td></tr>

<tr>

<td>(225)2 = 250 = 1125899906842624

</td></tr>

<tr>

<td>(250)2 = 2100 = 1267650600228229401496703205376

</td></tr></tbody></table>

This series of steps only requires 8 multiplications instead of 99.

In general, the number of multiplication operations required to compute bn can be reduced to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sharp n+\lfloor \log _{2}n\rfloor -1,}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sharp n+\lfloor \log _{2}n\rfloor -1,}</annotation>

</semantics>

</math><img src="./e4823a4a6d51e227dbcfecd32b4e1dcfe8ecd394.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.662ex; height:2.843ex;" alt="{\displaystyle \sharp n+\lfloor \log _{2}n\rfloor -1,}" loading="lazy"> by using <a href="Exponentiation_by_squaring" title="Exponentiation by squaring">exponentiation by squaring</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sharp n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sharp n}</annotation>

</semantics>

</math><img src="./248676a199694aa132839264e8ab39e490b1116e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.299ex; height:2.843ex;" alt="{\displaystyle \sharp n}" loading="lazy"> denotes the number of 1s in the <a href="Binary_representation" class="mw-redirect" title="Binary representation">binary representation</a> of n. For some exponents (100 is not among them), the number of multiplications can be further reduced by computing and using the minimal <a href="Addition-chain_exponentiation" title="Addition-chain exponentiation">addition-chain exponentiation</a>. Finding the minimal sequence of multiplications (the minimal-length addition chain for the exponent) for bn is a difficult problem, for which no efficient algorithms are currently known (see <a href="Subset_sum_problem" title="Subset sum problem">Subset sum problem</a>), but many reasonably efficient heuristic algorithms are available.<a href="#cite_note-46">[44]</a> However, in practical computations, exponentiation by squaring is efficient enough, and much more easy to implement.

<div class="mw-heading mw-heading2"><h2 id="Iterated_functions">Iterated functions</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Iterated_function" title="Iterated function">Iterated function</a></div>

<a href="Function_composition" title="Function composition">Function composition</a> is a <a href="Binary_operation" title="Binary operation">binary operation</a> that is defined on <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> such that the <a href="Codomain" title="Codomain">codomain</a> of the function written on the right is included in the <a href="Domain_of_a_function" title="Domain of a function">domain</a> of the function written on the left. It is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ f,}</annotation>

</semantics>

</math><img src="./3ce09bd3a001a295c1391297f3e3005d63f932e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.236ex; height:2.509ex;" alt="{\displaystyle g\circ f,}" loading="lazy"> and defined as

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (g\circ f)(x)=g(f(x))}</annotation>

</semantics>

</math><img src="./c8a73f8d834a602ee506ac323b8a36ce17ac2b9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.979ex; height:2.843ex;" alt="{\displaystyle (g\circ f)(x)=g(f(x))}" loading="lazy"></dd></dl>

for every x in the domain of f.

If the domain of a function f equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the nth power of the function under composition, commonly called the nth iterate of the function. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{n}}</annotation>

</semantics>

</math><img src="./2af7f3f2e54e4e35e26230a4e131c45e5d4000c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.539ex; height:2.676ex;" alt="{\displaystyle f^{n}}" loading="lazy"> denotes generally the nth iterate of f; for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{3}(x)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{3}(x)}</annotation>

</semantics>

</math><img src="./218b0ffd82ef0f3907e785313ec63131d5fb2bf1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.514ex; height:3.176ex;" alt="{\displaystyle f^{3}(x)}" loading="lazy"> means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(f(f(x))).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(f(f(x))).}</annotation>

</semantics>

</math><img src="./6a34894417ef1a9f43916fbe850bcdd24a405b54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.24ex; height:2.843ex;" alt="{\displaystyle f(f(f(x))).}" loading="lazy"><a href="#cite_note-Peano_1903-47">[45]</a>

When a multiplication is defined on the codomain of the function, this defines a multiplication on functions, the <a href="Pointwise_multiplication" class="mw-redirect" title="Pointwise multiplication">pointwise multiplication</a>, which induces another exponentiation. When using <a href="Functional_notation" class="mw-redirect" title="Functional notation">functional notation</a>, the two kinds of exponentiation are generally distinguished by placing the exponent of the functional iteration before the parentheses enclosing the arguments of the function, and placing the exponent of pointwise multiplication after the parentheses. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{2}(x)=f(f(x)),}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{2}(x)=f(f(x)),}</annotation>

</semantics>

</math><img src="./168e1d1303139726fcdfd9fa3f6621fc27b93afb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.764ex; height:3.176ex;" alt="{\displaystyle f^{2}(x)=f(f(x)),}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)^{2}=f(x)\cdot f(x).}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)^{2}=f(x)\cdot f(x).}</annotation>

</semantics>

</math><img src="./b094f70e2cbd87e83faf291e3b33ce4f95f6c43d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.731ex; height:3.176ex;" alt="{\displaystyle f(x)^{2}=f(x)\cdot f(x).}" loading="lazy"> When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{\circ 3}=f\circ f\circ f,}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{\circ 3}=f\circ f\circ f,}</annotation>

</semantics>

</math><img src="./f0bd923e0c297c2d816cd67435872ae24f84ca22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.167ex; height:3.009ex;" alt="{\displaystyle f^{\circ 3}=f\circ f\circ f,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{3}=f\cdot f\cdot f.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{3}=f\cdot f\cdot f.}</annotation>

</semantics>

</math><img src="./34267b672d6eb8d2dbfe5863843d8a5ebd1f956d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.314ex; height:3.009ex;" alt="{\displaystyle f^{3}=f\cdot f\cdot f.}" loading="lazy"> For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the <a href="Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>. So, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}x}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin ^{2}x}</annotation>

</semantics>

</math><img src="./fc0d6ba6bb181219b776ab25be991303f9e07d0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.627ex; height:2.676ex;" alt="{\displaystyle \sin ^{2}x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}(x)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin ^{2}(x)}</annotation>

</semantics>

</math><img src="./53a0a5d7313c2ee4d060de4479eb4d418ce73310.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.049ex; height:3.176ex;" alt="{\displaystyle \sin ^{2}(x)}" loading="lazy"> both mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x)\cdot \sin(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(x)\cdot \sin(x)}</annotation>

</semantics>

</math><img src="./c6abd8bbecb1c69b243659fadd4125845de237b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.668ex; height:2.843ex;" alt="{\displaystyle \sin(x)\cdot \sin(x)}" loading="lazy"> and not <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\sin(x)),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(\sin(x)),}</annotation>

</semantics>

</math><img src="./7f00134607d0fd014db3c83ee65d56efb531b379.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.306ex; height:2.843ex;" alt="{\displaystyle \sin(\sin(x)),}" loading="lazy"> which, in any case, is rarely considered. Historically, several variants of these notations were used by different authors.<a href="#cite_note-Herschel_1813-48">[46]</a><a href="#cite_note-Herschel_1820-49">[47]</a><a href="#cite_note-Cajori_1929-50">[48]</a>

In this context, the exponent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

</semantics>

</math><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy"> denotes always the <a href="Inverse_function" title="Inverse function">inverse function</a>, if it exists. So <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{-1}x=\sin ^{-1}(x)=\arcsin x.}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mi>arcsin</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin ^{-1}x=\sin ^{-1}(x)=\arcsin x.}</annotation>

</semantics>

</math><img src="./69491ab235891e82f885b351b52237efff1504d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.755ex; height:3.176ex;" alt="{\displaystyle \sin ^{-1}x=\sin ^{-1}(x)=\arcsin x.}" loading="lazy"> For the <a href="Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> fractions are generally used as in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/\sin(x)={\frac {1}{\sin x}}.}">

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/\sin(x)={\frac {1}{\sin x}}.}</annotation>

</semantics>

</math><img src="./504f40ff64eff66ed3eeded3869fde80a05496af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.86ex; height:5.176ex;" alt="{\displaystyle 1/\sin(x)={\frac {1}{\sin x}}.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="In_programming_languages">In programming languages</h2></div>

<a href="Programming_language" title="Programming language">Programming languages</a> generally express exponentiation either as an infix <a href="Operator_(computer_programming)" title="Operator (computer programming)">operator</a> or as a function application, as they do not support superscripts. The most common operator symbol for exponentiation is the <a href="Caret" title="Caret">caret</a> (<code>^</code>). The <a href="ASCII#1963" title="ASCII">original version of ASCII</a> included an uparrow symbol (<code>↑</code>), intended for exponentiation, but this was <a href="Caret#History" title="Caret">replaced by the caret</a> in 1967, so the caret became usual in programming languages.<a href="#cite_note-51">[49]</a>

The notations include:

<ul><li><code>x ^ y</code>: <a href="AWK" title="AWK">AWK</a>, <a href="BASIC" title="BASIC">BASIC</a>, <a href="J_programming_language" class="mw-redirect" title="J programming language">J</a>, <a href="MATLAB" title="MATLAB">MATLAB</a>, <a href="Wolfram_Language" title="Wolfram Language">Wolfram Language</a> (<a href="Wolfram_Mathematica" class="mw-redirect" title="Wolfram Mathematica">Mathematica</a>), <a href="R_(programming_language)" title="R (programming language)">R</a>, <a href="Microsoft_Excel" title="Microsoft Excel">Microsoft Excel</a>, <a href="Analytica_(software)" title="Analytica (software)">Analytica</a>, <a href="TeX" title="TeX">TeX</a> (and its derivatives), <a href="TI-BASIC" title="TI-BASIC">TI-BASIC</a>, <a href="Bc_programming_language" class="mw-redirect" title="Bc programming language">bc</a> (for integer exponents), <a href="Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell</a> (for nonnegative integer exponents), <a href="Lua_(programming_language)" class="mw-redirect" title="Lua (programming language)">Lua</a>, and most <a href="Computer_algebra_system" title="Computer algebra system">computer algebra systems</a>.</li>

<li><code>x ** y</code>. The <a href="Fortran" title="Fortran">Fortran</a> character set did not include lowercase characters or punctuation symbols other than <code>+-*/()&=.,'</code> and so used <code>**</code> for exponentiation<a href="#cite_note-Sayre_1956-52">[50]</a><a href="#cite_note-53">[51]</a> (the initial version used <code>a xx b</code> instead.<a href="#cite_note-Backus_1954-54">[52]</a>). Many other languages followed suit: <a href="Ada_(programming_language)" title="Ada (programming language)">Ada</a>, <a href="Z_shell" title="Z shell">Z shell</a>, <a href="KornShell" title="KornShell">KornShell</a>, <a href="Bash_(Unix_shell)" title="Bash (Unix shell)">Bash</a>, <a href="COBOL" title="COBOL">COBOL</a>, <a href="CoffeeScript" title="CoffeeScript">CoffeeScript</a>, <a href="Fortran" title="Fortran">Fortran</a>, <a href="FoxPro_2" class="mw-redirect" title="FoxPro 2">FoxPro</a>, <a href="Gnuplot" title="Gnuplot">Gnuplot</a>, <a href="Apache_Groovy" title="Apache Groovy">Groovy</a>, <a href="JavaScript" title="JavaScript">JavaScript</a>, <a href="OCaml" title="OCaml">OCaml</a>, <a href="Object_REXX" title="Object REXX">ooRexx</a>, <a href="F_Sharp_(programming_language)" title="F Sharp (programming language)">F#</a>, <a href="Perl" title="Perl">Perl</a>, <a href="PHP" title="PHP">PHP</a>, <a href="PL/I" title="PL/I">PL/I</a>, <a href="Python_(programming_language)" title="Python (programming language)">Python</a>, <a href="Rexx" title="Rexx">Rexx</a>, <a href="Ruby_(programming_language)" title="Ruby (programming language)">Ruby</a>, <a href="SAS_programming_language" class="mw-redirect" title="SAS programming language">SAS</a>, <a href="Seed7" title="Seed7">Seed7</a>, <a href="Tcl" title="Tcl">Tcl</a>, <a href="ABAP" title="ABAP">ABAP</a>, <a href="Mercury_(programming_language)" title="Mercury (programming language)">Mercury</a>, Haskell (for floating-point exponents), <a href="Turing_(programming_language)" title="Turing (programming language)">Turing</a>, and <a href="VHDL" title="VHDL">VHDL</a>.</li>

<li><code>x ↑ y</code>: <a href="Algol_programming_language" class="mw-redirect" title="Algol programming language">Algol Reference language</a>, <a href="Commodore_BASIC" title="Commodore BASIC">Commodore BASIC</a>, <a href="TRS-80_Level_II_BASIC" class="mw-redirect" title="TRS-80 Level II BASIC">TRS-80 Level II/III BASIC</a>.<a href="#cite_note-InfoWorld_1982-55">[53]</a><a href="#cite_note-80Micro_1983-56">[54]</a></li>

<li><code>x ^^ y</code>: Haskell (for fractional base, integer exponents), <a href="D_(programming_language)" title="D (programming language)">D</a>.</li>

In most programming languages with an infix exponentiation operator, it is <a href="Operator_associativity" title="Operator associativity">right-associative</a>, that is, <code>a^b^c</code> is interpreted as <code>a^(b^c)</code>.<a href="#cite_note-57">[55]</a> This is because <code>(a^b)^c</code> is equal to <code>a^(b*c)</code> and thus not as useful. In some languages, it is left-associative, notably in <a href="Algol" title="Algol">Algol</a>, <a href="MATLAB" title="MATLAB">MATLAB</a>, and the <a href="Microsoft_Office_Excel" class="mw-redirect" title="Microsoft Office Excel">Microsoft Excel</a> formula language.

Other programming languages use functional notation:

<ul><li><code>(expt x y)</code>: <a href="Common_Lisp" title="Common Lisp">Common Lisp</a>.</li>

<li><code>pown x y</code>: <a href="F_Sharp_(programming_language)" title="F Sharp (programming language)">F#</a> (for integer base, integer exponent).</li></ul>

Still others only provide exponentiation as part of standard <a href="Library_(computing)" title="Library (computing)">libraries</a>:

<ul><li><code>pow(x, y)</code>: <a href="C_(programming_language)" title="C (programming language)">C</a>, <a href="C%2B%2B" title="C++">C++</a> (in <code>math</code> library).</li>

<li><code>math:pow(X, Y)</code>: <a href="Erlang_(programming_language)" title="Erlang (programming language)">Erlang</a>.</li>

<li><code>[Math]::Pow(x, y)</code>: <a href="PowerShell" title="PowerShell">PowerShell</a>.</li></ul>

In some <a href="Type_system" title="Type system">statically typed</a> languages that prioritize <a href="Type_safety" title="Type safety">type safety</a> such as <a href="Rust_(programming_language)" title="Rust (programming language)">Rust</a>, exponentiation is performed via a multitude of methods:

<ul><li><code>x.pow(y)</code> for <code>x</code> and <code>y</code> as integers</li>

<li><code>x.powf(y)</code> for <code>x</code> and <code>y</code> as floating-point numbers</li>

<li><code>x.powi(y)</code> for <code>x</code> as a float and <code>y</code> as an integer</li></ul>

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<ul><li><a href="Double_exponential_function" title="Double exponential function">Double exponential function</a> – Exponential function of an exponential function</li>

<li><a href="Exponential_decay" title="Exponential decay">Exponential decay</a> – Decrease in value at a rate proportional to the current value</li>

<li><a href="Exponential_field" title="Exponential field">Exponential field</a> – Mathematical field with an extra operation</li>

<li><a href="Exponential_growth" title="Exponential growth">Exponential growth</a> – Growth of quantities at rate proportional to the current amount</li>

<li><a href="Pentation" title="Pentation">Pentation</a> – Arithmetic operation</li>

<li><a href="List_of_exponential_topics" title="List of exponential topics">List of exponential topics</a></li>

<li><a href="Modular_exponentiation" title="Modular exponentiation">Modular exponentiation</a> – Exponentation in modular arithmetic</li>

<li><a href="Unicode_subscripts_and_superscripts" title="Unicode subscripts and superscripts">Unicode subscripts and superscripts</a> – Unicode denominator & numerator glyphs</li>

<li><a href="Equation_x%5Ey_%3D_y%5Ex" class="mw-redirect" title="Equation x^y = y^x">xy = yx</a> – In general, exponentiation fails to be commutativePages displaying short descriptions of redirect targets</li></ul></div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> There are three common notations for <a href="Multiplication" title="Multiplication">multiplication</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\times y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\times y}</annotation>

</semantics>

</math><img src="./ec85d07724b6213a6af69367b22f33502f44dc4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.326ex; height:2.009ex;" alt="{\displaystyle x\times y}" loading="lazy"> is most commonly used for explicit numbers and at a very elementary level; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle xy}</annotation>

</semantics>

</math><img src="./c72eb345e496513fb8b2fa4aa8c4d89b855f9a01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}" loading="lazy"> is most common when <a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a> are used; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\cdot y}</annotation>

</semantics>

</math><img src="./13939e6cddd7ba416fd805830d8f5d815c9b4e76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.164ex; height:2.009ex;" alt="{\displaystyle x\cdot y}" loading="lazy"> is used for emphasizing that one talks of multiplication or when omitting the multiplication sign would be confusing.

</li>

<li id="cite_note-39"><a href="#cite_ref-39">^</a> More generally, <a href="Power_associativity" title="Power associativity">power associativity</a> is sufficient for the definition.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFNykamp" class="citation web cs1">Nykamp, Duane. <a rel="nofollow" class="external text" href="https://mathinsight.org/exponentiation_basic_rules">"Basic rules for exponentiation"</a>. Math Insight. Retrieved 2020-08-27.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Power.html">"Power"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>. Retrieved 2020-08-27.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.etymonline.com/word/exponent">"Exponent | Etymology of exponent by etymonline"</a>.</cite>

</li>

<li id="cite_note-Rotman-5">^ <a href="#cite_ref-Rotman_5-0">a</a> <a href="#cite_ref-Rotman_5-1">b</a> <cite id="CITEREFRotman2015" class="citation book cs1"><a href="Joseph_J._Rotman" title="Joseph J. Rotman">Rotman, Joseph J.</a> (2015). <a rel="nofollow" class="external text" href="https://www.ams.org/books/gsm/165/04">Advanced Modern Algebra, Part 1</a>. <a href="Graduate_Studies_in_Mathematics" title="Graduate Studies in Mathematics">Graduate Studies in Mathematics</a>. Vol. 165 (3rd ed.). Providence, RI: <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. p. 130, fn. 4. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4704-1554-9</bdi>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFSzabó1978" class="citation book cs1">Szabó, Árpád (1978). <a rel="nofollow" class="external text" href="https://archive.org/details/TheBeginningsOfGreekMathematics">The Beginnings of Greek Mathematics</a>. Synthese Historical Library. Vol. 17. Translated by A.M. Ungar. Dordrecht: <a href="D._Reidel" title="D. Reidel">D. Reidel</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/TheBeginningsOfGreekMathematics/page/n37">37</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>90-277-0819-3</bdi>.</cite>

</li>

<li id="cite_note-MacTutor-7">^ <a href="#cite_ref-MacTutor_7-0">a</a> <a href="#cite_ref-MacTutor_7-1">b</a> <cite id="CITEREFO'ConnorRobertson" class="citation cs1">O'Connor, John J.; <a href="Edmund_F._Robertson" class="mw-redirect" title="Edmund F. Robertson">Robertson, Edmund F.</a> <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Miscellaneous/Mathematical_notation.html">"Etymology of some common mathematical terms"</a>. <a href="MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a>. <a href="University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFBall1915" class="citation book cs1"><a href="W._W._Rouse_Ball" title="W. W. Rouse Ball">Ball, W. W. Rouse</a> (1915). <a rel="nofollow" class="external text" href="https://archive.org/details/shortaccountofhi00ballrich">A Short Account of the History of Mathematics</a> (6th ed.). London: <a href="Macmillan_Publishers" title="Macmillan Publishers">Macmillan</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/shortaccountofhi00ballrich/page/38">38</a>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20240915223031/https://jeff560.tripod.com/e.html">"Earliest Known Uses of Some of the Words of Mathematics (E)"</a>. 2017-06-23. Archived from <a rel="nofollow" class="external text" href="https://jeff560.tripod.com/e.html">the original</a> on 2024-09-15.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFStifel1544" class="citation book cs1"><a href="Michael_Stifel" title="Michael Stifel">Stifel, Michael</a> (1544). <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_fndPsRv08R0C/page/n491">Arithmetica integra</a>. Nuremberg: <a href="Johannes_Petreius" title="Johannes Petreius">Johannes Petreius</a>. p. 235v.</cite>

</li>

<li id="cite_note-worldwidewords-11">^ <a href="#cite_ref-worldwidewords_11-0">a</a> <a href="#cite_ref-worldwidewords_11-1">b</a> <cite id="CITEREFQuinion" class="citation web cs1"><a href="Michael_Quinion" title="Michael Quinion">Quinion, Michael</a>. <a rel="nofollow" class="external text" href="https://www.worldwidewords.org/weirdwords/ww-zen1.htm">"Zenzizenzizenzic"</a>. World Wide Words. Retrieved 2020-04-16.</cite>

</li>

Archimedes. (2009). THE SAND-RECKONER. In T. Heath (Ed.), The Works of Archimedes: Edited in Modern Notation with Introductory Chapters (Cambridge Library Collection - Mathematics, pp. 229-232). Cambridge: Cambridge University Press. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511695124.017">10.1017/CBO9780511695124.017</a>.

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFO'ConnorRobertson" class="citation cs1">O'Connor, John J.; <a href="Edmund_F._Robertson" class="mw-redirect" title="Edmund F. Robertson">Robertson, Edmund F.</a> <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Al-Qalasadi.html">"Abu'l Hasan ibn Ali al Qalasadi"</a>. <a href="MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a>. <a href="University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a>.</cite>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFCajori1928" class="citation book cs1">Cajori, Florian (1928). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema031756mbp">A History of Mathematical Notations</a>. Vol. 1. The Open Court Company. p. 102.</cite>

</li>

<li id="cite_note-cajori-15"><a href="#cite_ref-cajori_15-0">^</a> <cite id="CITEREFCajori1928" class="citation book cs1"><a href="Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1928). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema031756mbp">A History of Mathematical Notations</a>. Vol. 1. London: <a href="Open_Court_Publishing_Company" title="Open Court Publishing Company">Open Court Publishing Company</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema031756mbp/page/n363">344</a>.</cite>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFCajori1928" class="citation book cs1">Cajori, Florian (1928). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema031756mbp">A History of Mathematical Notations</a>. Vol. 1. The Open Court Company. p. 204.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFDescartes1637" class="citation book cs1"><a href="Ren%C3%A9_Descartes" title="René Descartes">Descartes, René</a> (1637). "<a href="La_G%C3%A9om%C3%A9trie" title="La Géométrie">La Géométrie</a>". <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/btv1b86069594/f383.image">Discourse de la méthode [...]</a>. Leiden: Jan Maire. p. 299. <q>Et aa, ou a2, pour multiplier a par soy mesme; Et a3, pour le multiplier encore une fois par a, & ainsi a l'infini</q></cite> (And aa, or a2, in order to multiply a by itself; and a3, in order to multiply it once more by a, and thus to infinity).

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> The most recent usage in this sense cited by the OED is from 1806 (<cite id="CITEREFReference-OED-involution" class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.oed.com/search/dictionary/?q=involution">"involution"</a>. <a href="Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a> (Online ed.). <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</cite> (Subscription or <a rel="nofollow" class="external text" href="https://www.oed.com/public/login/loggingin#withyourlibrary">participating institution membership</a> required.)).

</li>

<li id="cite_note-Euler_1748-19"><a href="#cite_ref-Euler_1748_19-0">^</a> <cite id="CITEREFEuler1748" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="Leonhard_Euler" title="Leonhard Euler">Euler, Leonhard</a> (1748). <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k33510/f93.image">Introductio in analysin infinitorum</a> (in Latin). Vol. I. Lausanne: Marc-Michel Bousquet. pp. 69, 98–99. <q>Primum ergo considerandæ sunt quantitates exponentiales, seu Potestates, quarum Exponens ipse est quantitas variabilis. Perspicuum enim est hujusmodi quantitates ad Functiones algebraicas referri non posse, cum in his Exponentes non nisi constantes locum habeant.</q></cite>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> Janet Shiver & Terri Wiilard "<a rel="nofollow" class="external text" href="https://www.visionlearning.com/en/library/Math-in-Science/62/Scientific-Notation/250">Scientific notation: working with orders of magnitude</a> from <a href="Visionlearning" title="Visionlearning">Visionlearning</a>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> School Mathematics Study Group (1961) Mathematics for Junior High School, volume 2, part 1, <a href="Yale_University_Press" title="Yale University Press">Yale University Press</a>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> Cecelia Callanan (1967) "Scientific Notation", <a href="The_Mathematics_Teacher" class="mw-redirect" title="The Mathematics Teacher">The Mathematics Teacher</a> 60: 252–6 <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27957540">JSTOR</a>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <a href="Edwin_Bidwell_Wilson" title="Edwin Bidwell Wilson">Edwin Bidwell Wilson</a> (1920) <a rel="nofollow" class="external text" href="https://archive.org/details/aeronauticsclass00wilsrich/page/182/mode/2up">Theory of Dimensions</a>, chapter 11 in Aeronautics: A Class Text, via Internet Archive

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFBridgman1922" class="citation book cs1"><a href="Percy_Bridgman" class="mw-redirect" title="Percy Bridgman">Bridgman, Percy Williams</a> (1922). <a rel="nofollow" class="external text" href="https://archive.org/details/dimensionalanaly00bridrich">Dimensional Analysis</a>. New Haven: Yale University Press. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/840631">840631</a>.</cite>

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite id="CITEREFHodgeSchlickerSundstorm2014" class="citation book cs1">Hodge, Jonathan K.; Schlicker, Steven; Sundstorm, Ted (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qToTAgAAQBAJ&pg=PA94">Abstract Algebra: an inquiry based approach</a>. CRC Press. p. 94. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4665-6706-1</bdi>.</cite>

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite id="CITEREFAchatz2005" class="citation book cs1">Achatz, Thomas (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YOdtemSmzQQC&pg=PA101">Technical Shop Mathematics</a> (3rd ed.). Industrial Press. p. 101. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8311-3086-2</bdi>.</cite>

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> <cite id="CITEREFKnobloch1994" class="citation book cs1"><a href="Eberhard_Knobloch" title="Eberhard Knobloch">Knobloch, Eberhard</a> (1994). "The infinite in Leibniz's mathematics – The historiographical method of comprehension in context". In Kostas Gavroglu; Jean Christianidis; Efthymios Nicolaidis (eds.). Trends in the Historiography of Science. Boston Studies in the Philosophy of Science. Vol. 151. Springer Netherlands. p. 276. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-017-3596-4_20">10.1007/978-94-017-3596-4_20</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9789401735964</bdi>. <q>A positive power of zero is infinitely small, a negative power of zero is infinite.</q></cite>

</li>

<li id="cite_note-Bronstein_1987-28"><a href="#cite_ref-Bronstein_1987_28-0">^</a> <cite id="CITEREFBronsteinSemendjajew1987" class="citation book cs1 cs1-prop-location-test cs1-prop-foreign-lang-source"><a href="Ilya_Nikolaevich_Bronshtein" title="Ilya Nikolaevich Bronshtein">Bronstein, Ilja Nikolaevič</a>; <a href="Konstantin_Adolfovic_Semendyayev" class="mw-redirect" title="Konstantin Adolfovic Semendyayev">Semendjajew, Konstantin Adolfovič</a> (1987) [1945]. "2.4.1.1. Definition arithmetischer Ausdrücke" [Definition of arithmetic expressions]. Written at Leipzig, Germany. In Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea (eds.). <a href="Bronstein_and_Semendjajew" class="mw-redirect" title="Bronstein and Semendjajew">Taschenbuch der Mathematik</a> [Pocketbook of mathematics] (in German). Vol. 1. Translated by Ziegler, Viktor. Weiß, Jürgen (23 ed.). Thun, Switzerland / Frankfurt am Main, Germany: <a href="Verlag_Harri_Deutsch" title="Verlag Harri Deutsch">Verlag Harri Deutsch</a> (and <a href="B._G._Teubner_Verlagsgesellschaft" class="mw-redirect" title="B. G. Teubner Verlagsgesellschaft">B. G. Teubner Verlagsgesellschaft</a>, Leipzig). pp. 115–120, 802. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-87144-492-8</bdi>.</cite>

</li>

<li id="cite_note-NIST_2010-29"><a href="#cite_ref-NIST_2010_29-0">^</a> <cite id="CITEREFOlverLozierBoisvertClark2010" class="citation book cs1">Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010). <a href="NIST_Handbook_of_Mathematical_Functions" class="mw-redirect" title="NIST Handbook of Mathematical Functions">NIST Handbook of Mathematical Functions</a>. <a href="National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a> (NIST), <a href="U.S._Department_of_Commerce" class="mw-redirect" title="U.S. Department of Commerce">U.S. Department of Commerce</a>, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19225-5</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a>.</cite><a rel="nofollow" class="external autonumber" href="http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521140638">[1]</a>

</li>

<li id="cite_note-Zeidler_2013-30"><a href="#cite_ref-Zeidler_2013_30-0">^</a> <cite id="CITEREFZeidlerSchwarzHackbuschLuderer2013" class="citation book cs1 cs1-prop-interwiki-linked-name cs1-prop-foreign-lang-source"><a href="https://de.wikipedia.org/wiki/Eberhard_Zeidler" class="extiw external" title="de:Eberhard Zeidler">Zeidler, Eberhard</a> [in German]; Schwarz, Hans Rudolf; <a href="Wolfgang_Hackbusch" title="Wolfgang Hackbusch">Hackbusch, Wolfgang</a>; <a href="https://de.wikipedia.org/wiki/Bernd_Luderer" class="extiw external" title="de:Bernd Luderer">Luderer, Bernd</a> [in German]; Blath, Jochen; Schied, Alexander; Dempe, Stephan; <a href="Gert_Wanka" class="mw-redirect" title="Gert Wanka">Wanka, Gert</a>; <a href="Juraj_Hromkovi%C4%8D" title="Juraj Hromkovič">Hromkovič, Juraj</a>; <a href="Siegfried_Gottwald" title="Siegfried Gottwald">Gottwald, Siegfried</a> (2013) [2012]. <a href="https://de.wikipedia.org/wiki/Eberhard_Zeidler" class="extiw external" title="de:Eberhard Zeidler">Zeidler, Eberhard</a> [in German] (ed.). <a href="Springer-Handbuch_der_Mathematik" class="mw-redirect" title="Springer-Handbuch der Mathematik">Springer-Handbuch der Mathematik I</a> (in German). Vol. I (1 ed.). Berlin / Heidelberg, Germany: <a href="Springer_Spektrum" class="mw-redirect" title="Springer Spektrum">Springer Spektrum</a>, <a href="Springer_Fachmedien_Wiesbaden" class="mw-redirect" title="Springer Fachmedien Wiesbaden">Springer Fachmedien Wiesbaden</a>. p. 590. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-658-00284-8</bdi>.</cite> (xii+635 pages)

</li>

<li id="cite_note-31"><a href="#cite_ref-31">^</a> <cite id="CITEREFHassHeilWeirThomas2018" class="citation book cs1">Hass, Joel R.; Heil, Christopher E.; Weir, Maurice D.; Thomas, George B. (2018). Thomas' Calculus (14 ed.). Pearson. pp. 7–8. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780134439020</bdi>.</cite>

</li>

<li id="cite_note-Calculus:_Early_Transcendentals-32">^ <a href="#cite_ref-Calculus:_Early_Transcendentals_32-0">a</a> <a href="#cite_ref-Calculus:_Early_Transcendentals_32-1">b</a> <cite id="CITEREFAntonBivensDavis2012" class="citation book cs1">Anton, Howard; Bivens, Irl; Davis, Stephen (2012). <a rel="nofollow" class="external text" href="https://archive.org/details/calculusearlytra00anto_656">Calculus: Early Transcendentals</a> (9th ed.). John Wiley & Sons. p. <a rel="nofollow" class="external text" href="https://archive.org/details/calculusearlytra00anto_656/page/n51">28</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780470647691</bdi>.</cite>

</li>

<li id="cite_note-Denlinger-33"><a href="#cite_ref-Denlinger_33-0">^</a> <cite id="CITEREFDenlinger2011" class="citation book cs1">Denlinger, Charles G. (2011). Elements of Real Analysis. Jones and Bartlett. pp. 278–283. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7637-7947-4</bdi>.</cite>

</li>

<li id="cite_note-34"><a href="#cite_ref-34">^</a> <cite id="CITEREFTao2016" class="citation book cs1">Tao, Terence (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ecTsDAAAQBAJ&pg=PA154">"Limits of sequences"</a>. Analysis I. Texts and Readings in Mathematics. Vol. 37. pp. 126–154. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-981-10-1789-6_6">10.1007/978-981-10-1789-6_6</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-10-1789-6</bdi>.</cite>

</li>

<li id="cite_note-35"><a href="#cite_ref-35">^</a> <cite id="CITEREFCormenLeisersonRivestStein2001" class="citation book cs1">Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). Introduction to Algorithms (second ed.). <a href="MIT_Press" title="MIT Press">MIT Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-262-03293-3</bdi>.</cite> <a rel="nofollow" class="external text" href="http://highered.mcgraw-hill.com/sites/0070131511/student_view0/chapter30/glossary.html">Online resource</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070930201902/http://highered.mcgraw-hill.com/sites/0070131511/student_view0/chapter30/glossary.html">Archived</a> 2007-09-30 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.

</li>

<li id="cite_note-36"><a href="#cite_ref-36">^</a> <cite id="CITEREFCullFlahiveRobson2005" class="citation book cs1">Cull, Paul; <a href="Mary_Flahive" title="Mary Flahive">Flahive, Mary</a>; Robson, Robby (2005). <a href="Difference_Equations%3A_From_Rabbits_to_Chaos" title="Difference Equations: From Rabbits to Chaos">Difference Equations: From Rabbits to Chaos</a> (<a href="Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a> ed.). Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-23234-8</bdi>.</cite> Defined on p. 351.

</li>

<li id="cite_note-37"><a href="#cite_ref-37">^</a> <cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PrincipalRootofUnity.html">"Principal root of unity"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite>

</li>

<li id="cite_note-Clausen1827-38"><a href="#cite_ref-Clausen1827_38-0">^</a> <cite id="CITEREFSteinerClausenAbel1827" class="citation journal cs1">Steiner, J.; Clausen, T.; <a href="Niels_Henrik_Abel" title="Niels Henrik Abel">Abel, Niels Henrik</a> (1827). <a rel="nofollow" class="external text" href="https://www.digizeitschriften.de/dms/img/?PID=PPN243919689_0002%7Clog33&physid=phys301#navi">"Aufgaben und Lehrsätze, erstere aufzulösen, letztere zu beweisen"</a> [Problems and propositions, the former to solve, the later to prove]. <a href="Crelle's_Journal" title="Crelle's Journal">Journal für die reine und angewandte Mathematik</a>. 2: 286–287.</cite>

</li>

<li id="cite_note-40"><a href="#cite_ref-40">^</a> <cite id="CITEREFBourbaki1970" class="citation book cs1">Bourbaki, Nicolas (1970). Algèbre. Springer. I.2.</cite>

</li>

<li id="cite_note-41"><a href="#cite_ref-41">^</a> <cite id="CITEREFBloom1979" class="citation book cs1">Bloom, David M. (1979). <a rel="nofollow" class="external text" href="https://archive.org/details/linearalgebrageo0000bloo">Linear Algebra and Geometry</a>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/linearalgebrageo0000bloo/page/45">45</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-29324-2</bdi>.</cite>

</li>

<li id="cite_note-42"><a href="#cite_ref-42">^</a> Chapter 1, Elementary Linear Algebra, 8E, Howard Anton.

</li>

<li id="cite_note-43"><a href="#cite_ref-43">^</a> <cite id="CITEREFStrang1988" class="citation cs1">Strang, Gilbert (1988). Linear algebra and its applications (3rd ed.). Brooks-Cole. Chapter 5.</cite>

</li>

<li id="cite_note-44"><a href="#cite_ref-44">^</a> E. Hille, R. S. Phillips: Functional Analysis and Semi-Groups. American Mathematical Society, 1975.

</li>

<li id="cite_note-45"><a href="#cite_ref-45">^</a> Nicolas Bourbaki, Topologie générale, V.4.2.

</li>

<li id="cite_note-46"><a href="#cite_ref-46">^</a> <cite id="CITEREFGordon1998" class="citation journal cs1">Gordon, D. M. (1998). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180723164121/http://www.ccrwest.org/gordon/jalg.pdf">"A Survey of Fast Exponentiation Methods"</a> (PDF). Journal of Algorithms. 27: 129–146. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.7076">10.1.1.17.7076</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjagm.1997.0913">10.1006/jagm.1997.0913</a>. Archived from <a rel="nofollow" class="external text" href="http://www.ccrwest.org/gordon/jalg.pdf">the original</a> (PDF) on 2018-07-23. Retrieved 2024-01-11.</cite>

</li>

<li id="cite_note-Peano_1903-47"><a href="#cite_ref-Peano_1903_47-0">^</a> <cite id="CITEREFPeano1903" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="Giuseppe_Peano" title="Giuseppe Peano">Peano, Giuseppe</a> (1903). Formulaire mathématique (in French). Vol. IV. p. 229.</cite>

</li>

<li id="cite_note-Herschel_1813-48"><a href="#cite_ref-Herschel_1813_48-0">^</a> <cite id="CITEREFHerschel1813" class="citation journal cs1"><a href="John_Frederick_William_Herschel" class="mw-redirect" title="John Frederick William Herschel">Herschel, John Frederick William</a> (1813) [1812-11-12]. "On a Remarkable Application of Cotes's Theorem". <a href="Philosophical_Transactions_of_the_Royal_Society_of_London" class="mw-redirect" title="Philosophical Transactions of the Royal Society of London">Philosophical Transactions of the Royal Society of London</a>. 103 (Part 1). London: <a href="Royal_Society_of_London" class="mw-redirect" title="Royal Society of London">Royal Society of London</a>, printed by W. Bulmer and Co., Cleveland-Row, St. James's, sold by G. and W. Nicol, Pall-Mall: 8–26 [10]. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1813.0005">10.1098/rstl.1813.0005</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/107384">107384</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118124706">118124706</a>.</cite>

</li>

<li id="cite_note-Herschel_1820-49"><a href="#cite_ref-Herschel_1820_49-0">^</a> <cite id="CITEREFHerschel1820" class="citation book cs1"><a href="John_Frederick_William_Herschel" class="mw-redirect" title="John Frederick William Herschel">Herschel, John Frederick William</a> (1820). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PWcSAAAAIAAJ&pg=PA5">"Part III. Section I. Examples of the Direct Method of Differences"</a>. A Collection of Examples of the Applications of the Calculus of Finite Differences. Cambridge, UK: Printed by J. Smith, sold by J. Deighton & sons. pp. 1–13 [5–6]. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200804031020/https://books.google.de/books?id=PWcSAAAAIAAJ&jtp=5">Archived</a> from the original on 2020-08-04. Retrieved 2020-08-04.</cite> <a rel="nofollow" class="external autonumber" href="https://archive.org/details/acollectionexam00lacrgoog">[2]</a> (NB. Inhere, Herschel refers to his <a href="#CITEREFHerschel1813">1813 work</a> and mentions <a href="Hans_Heinrich_B%C3%BCrmann" title="Hans Heinrich Bürmann">Hans Heinrich Bürmann</a>'s older work.)

</li>

<li id="cite_note-Cajori_1929-50"><a href="#cite_ref-Cajori_1929_50-0">^</a> <cite id="CITEREFCajori1952" class="citation book cs1"><a href="Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1952) [March 1929]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bT5suOONXlgC">A History of Mathematical Notations</a>. Vol. 2 (3rd ed.). Chicago, USA: <a href="Open_court_publishing_company" class="mw-redirect" title="Open court publishing company">Open court publishing company</a>. pp. 108, 176–179, 336, 346. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-60206-714-1</bdi>. Retrieved 2016-01-18.</cite> <code class="cs1-code">{{cite book}}</code>: ISBN / Date incompatibility (help)

</li>

<li id="cite_note-51"><a href="#cite_ref-51">^</a> <cite id="CITEREFRichard_Gillam2003" class="citation book cs1">Richard Gillam (2003). Unicode Demystified: A Practical Programmer's Guide to the Encoding Standard. Addison-Wesley Professional. p. 33. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0201700522</bdi>.</cite>

</li>

<li id="cite_note-Sayre_1956-52"><a href="#cite_ref-Sayre_1956_52-0">^</a> <cite id="CITEREFBackusBeeberBestGoldberg1956" class="citation book cs1"><a href="John_Warner_Backus" class="mw-redirect" title="John Warner Backus">Backus, John Warner</a>; Beeber, R. J.; Best, Sheldon F.; <a href="Richard_Goldberg" title="Richard Goldberg">Goldberg, Richard</a>; Herrick, Harlan L.; Hughes, R. A.; Mitchell, L. B.; Nelson, Robert A.; <a href="Roy_Nutt" title="Roy Nutt">Nutt, Roy</a>; <a href="David_Sayre" title="David Sayre">Sayre, David</a>; Sheridan, Peter B.; Stern, Harold; Ziller, Irving (1956-10-15). <a href="David_Sayre" title="David Sayre">Sayre, David</a> (ed.). <a rel="nofollow" class="external text" href="https://archive.computerhistory.org/resources/text/Fortran/102649787.05.01.acc.pdf">The FORTRAN Automatic Coding System for the IBM 704 EDPM: Programmer's Reference Manual</a> (PDF). New York, USA: Applied Science Division and Programming Research Department, <a href="International_Business_Machines_Corporation" class="mw-redirect" title="International Business Machines Corporation">International Business Machines Corporation</a>. p. 15. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220704193549/http://archive.computerhistory.org/resources/text/Fortran/102649787.05.01.acc.pdf">Archived</a> (PDF) from the original on 2022-07-04. Retrieved 2022-07-04.</cite> (2+51+1 pages)

</li>

<li id="cite_note-53"><a href="#cite_ref-53">^</a> <cite id="CITEREFBrice_CarnahanJames_O._Wilkes1968" class="citation cs1">Brice Carnahan; James O. Wilkes (1968). Introduction to Digital Computing and FORTRAN IV with MTS Applications. pp. 2–2, 2–6.</cite>

</li>

<li id="cite_note-Backus_1954-54"><a href="#cite_ref-Backus_1954_54-0">^</a> <cite id="CITEREFBackusHerrickNelsonZiller1954" class="citation book cs1"><a href="John_Warner_Backus" class="mw-redirect" title="John Warner Backus">Backus, John Warner</a>; Herrick, Harlan L.; Nelson, Robert A.; Ziller, Irving (1954-11-10). <a href="John_Warner_Backus" class="mw-redirect" title="John Warner Backus">Backus, John Warner</a> (ed.). <a rel="nofollow" class="external text" href="https://archive.computerhistory.org/resources/text/Fortran/102679231.05.01.acc.pdf">Specifications for: The IBM Mathematical FORmula TRANSlating System, FORTRAN</a> (PDF) (Preliminary report). New York, USA: Programming Research Group, Applied Science Division, <a href="International_Business_Machines_Corporation" class="mw-redirect" title="International Business Machines Corporation">International Business Machines Corporation</a>. pp. 4, 6. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220329234459/https://archive.computerhistory.org/resources/text/Fortran/102679231.05.01.acc.pdf">Archived</a> (PDF) from the original on 2022-03-29. Retrieved 2022-07-04.</cite> (29 pages)

</li>

<li id="cite_note-InfoWorld_1982-55"><a href="#cite_ref-InfoWorld_1982_55-0">^</a> <cite id="CITEREFDaneliuk1982" class="citation news cs1">Daneliuk, Timothy "Tim" A. (1982-08-09). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NDAEAAAAMBAJ&pg=PA42">"BASCOM - A BASIC compiler for TRS-80 I and II"</a>. <a href="InfoWorld" title="InfoWorld">InfoWorld</a>. Software Reviews. Vol. 4, no. 31. <a href="Popular_Computing%2C_Inc." class="mw-redirect" title="Popular Computing, Inc.">Popular Computing, Inc.</a> pp. 41–42. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200207104336/https://books.google.de/books?id=NDAEAAAAMBAJ&pg=PA42&focus=viewport#v=onepage&q=TRS-80%20exponention">Archived</a> from the original on 2020-02-07. Retrieved 2020-02-06.</cite>

</li>

<li id="cite_note-80Micro_1983-56"><a href="#cite_ref-80Micro_1983_56-0">^</a> <cite class="citation journal cs1"><a rel="nofollow" class="external text" href="https://archive.org/details/80-microcomputing-magazine-1983-10">"80 Contents"</a>. <a href="80_Micro" title="80 Micro">80 Micro</a> (45). <a href="1001001%2C_Inc." class="mw-redirect" title="1001001, Inc.">1001001, Inc.</a>: 5. October 1983. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0744-7868">0744-7868</a>. Retrieved 2020-02-06.</cite>

</li>

<li id="cite_note-57"><a href="#cite_ref-57">^</a> <cite id="CITEREFRobert_W._Sebesta2010" class="citation book cs1">Robert W. Sebesta (2010). Concepts of Programming Languages. Addison-Wesley. pp. 130, 324. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0136073475</bdi>.</cite>

</li>

</ol></div></div>

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<ul><li><a href="Successor_function" title="Successor function">Successor (0)</a></li>

<li><a href="Addition" title="Addition">Addition (1)</a></li>

<li><a href="Multiplication" title="Multiplication">Multiplication (2)</a></li>

<li><a href="Tetration" title="Tetration">Tetration (4)</a></li>

<li><a href="Pentation" title="Pentation">Pentation (5)</a></li>

<li><a href="Hexation" class="mw-redirect" title="Hexation">Hexation (6)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Inverse_function" title="Inverse function">Inverse</a> for left argument</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Primitive_recursive_function#Predecessor" title="Primitive recursive function">Predecessor (0)</a></li>

<li><a href="Subtraction" title="Subtraction">Subtraction (1)</a></li>

<li><a href="Division_(mathematics)" title="Division (mathematics)">Division (2)</a></li>

<li><a href="Nth_root" title="Nth root">Root extraction (3)</a></li>

<li><a href="Super-root" class="mw-redirect" title="Super-root">Super-root (4)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Inverse for right argument</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Primitive_recursive_function#Predecessor" title="Primitive recursive function">Predecessor (0)</a></li>

<li><a href="Subtraction" title="Subtraction">Subtraction (1)</a></li>

<li><a href="Division_(mathematics)" title="Division (mathematics)">Division (2)</a></li>

<li><a href="Logarithm" title="Logarithm">Logarithm (3)</a></li>

<li><a href="Super-logarithm" class="mw-redirect" title="Super-logarithm">Super-logarithm (4)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related articles</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ackermann_function" title="Ackermann function">Ackermann function</a></li>

<li><a href="Conway_chained_arrow_notation" title="Conway chained arrow notation">Conway chained arrow notation</a></li>

<li><a href="Grzegorczyk_hierarchy" title="Grzegorczyk hierarchy">Grzegorczyk hierarchy</a></li>

<li><a href="Knuth's_up-arrow_notation" title="Knuth's up-arrow notation">Knuth's up-arrow notation</a></li>

<li><a href="Steinhaus%E2%80%93Moser_notation" title="Steinhaus–Moser notation">Steinhaus–Moser notation</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Orders_of_magnitude_of_time123" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Orders_of_magnitude_of_time123" style="font-size:114%;margin:0 4em"><a href="Orders_of_magnitude_(time)" title="Orders of magnitude (time)">Orders of magnitude of time</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>by of ten</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Negative powers</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Attosecond" title="Attosecond">attosecond</a></li>

<li><a href="Femtosecond" title="Femtosecond">femtosecond</a></li>

<li><a href="Picosecond" title="Picosecond">picosecond</a></li>

<li><a href="Nanosecond" title="Nanosecond">nanosecond</a></li>

<li><a href="Microsecond" title="Microsecond">microsecond</a></li>

<li><a href="Millisecond" title="Millisecond">millisecond</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Positive powers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Second" title="Second">second</a></li>

<li><a href="Kilosecond" class="mw-redirect" title="Kilosecond">kilosecond</a></li>

<li><a href="Megasecond" class="mw-redirect" title="Megasecond">megasecond</a></li>

<li><a href="Gigasecond" class="mw-redirect" title="Gigasecond">gigasecond</a></li>

<li><a href="Terasecond_and_longer" class="mw-redirect" title="Terasecond and longer">terasecond and longer</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Classes_of_natural_numbers743" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Classes_of_natural_numbers743" style="font-size:114%;margin:0 4em">Classes of <a href="Natural_number" title="Natural number">natural numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Powers_and_related_numbers743" style="font-size:114%;margin:0 4em"> and related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Achilles_number" title="Achilles number">Achilles</a></li>

<li><a href="Power_of_two" title="Power of two">Power of 2</a></li>

<li><a href="Power_of_three" title="Power of three">Power of 3</a></li>

<li><a href="Power_of_10" title="Power of 10">Power of 10</a></li>

<li><a href="Square_number" title="Square number">Square</a></li>

<li><a href="Fourth_power" title="Fourth power">Fourth power</a></li>

<li><a href="Fifth_power_(algebra)" title="Fifth power (algebra)">Fifth power</a></li>

<li><a href="Sixth_power" title="Sixth power">Sixth power</a></li>

<li><a href="Seventh_power" title="Seventh power">Seventh power</a></li>

<li><a href="Eighth_power" title="Eighth power">Eighth power</a></li>

<li><a href="Perfect_power" title="Perfect power">Perfect power</a></li>

<li><a href="Powerful_number" title="Powerful number">Powerful</a></li>

<li><a href="Prime_power" title="Prime power">Prime power</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1743" style="font-size:114%;margin:0 4em">Of the form a × 2b ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cullen_number" title="Cullen number">Cullen</a></li>

<li><a href="Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li>

<li><a href="Fermat_number" title="Fermat number">Fermat</a></li>

<li><a href="Mersenne_prime" title="Mersenne prime">Mersenne</a></li>

<li><a href="Proth_number" class="mw-redirect" title="Proth number">Proth</a></li>

<li><a href="Thabit_number" title="Thabit number">Thabit</a></li>

<li><a href="Woodall_number" title="Woodall number">Woodall</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers743" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hilbert_number" title="Hilbert number">Hilbert</a></li>

<li><a href="Idoneal_number" title="Idoneal number">Idoneal</a></li>

<li><a href="Leyland_number" title="Leyland number">Leyland</a></li>

<li><a href="Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li>

<li><a href="Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers743" style="font-size:114%;margin:0 4em"><a href="Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li>

<li><a href="Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li>

<li><a href="Leonardo_number" title="Leonardo number">Leonardo</a></li>

<li><a href="Lucas_number" title="Lucas number">Lucas</a></li>

<li><a href="Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li>

<li><a href="Padovan_sequence" title="Padovan sequence">Padovan</a></li>

<li><a href="Perrin_number" title="Perrin number">Perrin</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers743" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amenable_number" title="Amenable number">Amenable</a></li>

<li><a href="Congruent_number" title="Congruent number">Congruent</a></li>

<li><a href="Kn%C3%B6del_number" title="Knödel number">Knödel</a></li>

<li><a href="Riesel_number" title="Riesel number">Riesel</a></li>

<li><a href="Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums743" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li>

<li><a href="Polite_number" title="Polite number">Polite</a></li>

<li><a href="Practical_number" title="Practical number">Practical</a></li>

<li><a href="Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li>

<li><a href="Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers743" style="font-size:114%;margin:0 4em"><a href="Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li>

<li><a href="Centered_square_number" title="Centered square number">Centered square</a></li>

<li><a href="Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li>

<li><a href="Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li>

<li><a href="Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li>

<li><a href="Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li>

<li><a href="Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li>

<li><a href="Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Triangular_number" title="Triangular number">Triangular</a></li>

<li><a href="Square_number" title="Square number">Square</a></li>

<li><a href="Square_triangular_number" title="Square triangular number">Square triangular</a></li>

<li><a href="Pentagonal_number" title="Pentagonal number">Pentagonal</a></li>

<li><a href="Hexagonal_number" title="Hexagonal number">Hexagonal</a></li>

<li><a href="Heptagonal_number" title="Heptagonal number">Heptagonal</a></li>

<li><a href="Octagonal_number" title="Octagonal number">Octagonal</a></li>

<li><a href="Nonagonal_number" title="Nonagonal number">Nonagonal</a></li>

<li><a href="Decagonal_number" title="Decagonal number">Decagonal</a></li>

<li><a href="Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li>

<li><a href="Centered_cube_number" title="Centered cube number">Centered cube</a></li>

<li><a href="Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li>

<li><a href="Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li>

<li><a href="Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li>

<li><a href="Cube_(algebra)" title="Cube (algebra)">Cubic</a></li>

<li><a href="Octahedral_number" title="Octahedral number">Octahedral</a></li>

<li><a href="Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li>

<li><a href="Icosahedral_number" title="Icosahedral number">Icosahedral</a></li>

<li><a href="Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pentatope_number" title="Pentatope number">Pentatope</a></li>

<li><a href="Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li>

<li><a href="Fourth_power" title="Fourth power">Tesseractic</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers743" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Catalan_number" title="Catalan number">Catalan</a></li>

<li><a href="Dedekind_number" title="Dedekind number">Dedekind</a></li>

<li><a href="Delannoy_number" title="Delannoy number">Delannoy</a></li>

<li><a href="Euler_number" class="mw-redirect" title="Euler number">Euler</a></li>

<li><a href="Eulerian_number" title="Eulerian number">Eulerian</a></li>

<li><a href="Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li>

<li><a href="Lazy_caterer's_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li>

<li><a href="Motzkin_number" title="Motzkin number">Motzkin</a></li>

<li><a href="Narayana_number" title="Narayana number">Narayana</a></li>

<li><a href="Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li>

<li><a href="Schr%C3%B6der_number" title="Schröder number">Schröder</a></li>

<li><a href="Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li>

<li><a href="Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling first</a></li>

<li><a href="Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li>

<li><a href="Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li>

<li><a href="Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes743" style="font-size:114%;margin:0 4em"><a href="Prime_number" title="Prime number">Primes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</a></li>

<li><a href="Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a></li>

<li><a href="Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes743" style="font-size:114%;margin:0 4em"><a href="Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Carmichael_number" title="Carmichael number">Carmichael number</a></li>

<li><a href="Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li>

<li><a href="Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li>

<li><a href="Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li>

<li><a href="Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li>

<li><a href="Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li>

<li><a href="Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li>

<li><a href="Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li>

<li><a href="Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li>

<li><a href="Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li>

<li><a href="Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li>

<li><a href="Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics743" style="font-size:114%;margin:0 4em"><a href="Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Divisor_function" title="Divisor function">Divisor functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abundant_number" title="Abundant number">Abundant</a></li>

<li><a href="Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li>

<li><a href="Arithmetic_number" title="Arithmetic number">Arithmetic</a></li>

<li><a href="Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li>

<li><a href="Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li>

<li><a href="Deficient_number" title="Deficient number">Deficient</a></li>

<li><a href="Descartes_number" title="Descartes number">Descartes</a></li>

<li><a href="Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li>

<li><a href="Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li>

<li><a href="Highly_composite_number" title="Highly composite number">Highly composite</a></li>

<li><a href="Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li>

<li><a href="Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li>

<li><a href="Perfect_number" title="Perfect number">Perfect</a></li>

<li><a href="Practical_number" title="Practical number">Practical</a></li>

<li><a href="Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li>

<li><a href="Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li>

<li><a href="Refactorable_number" title="Refactorable number">Refactorable</a></li>

<li><a href="Semiperfect_number" title="Semiperfect number">Semiperfect</a></li>

<li><a href="Sublime_number" title="Sublime number">Sublime</a></li>

<li><a href="Superabundant_number" title="Superabundant number">Superabundant</a></li>

<li><a href="Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li>

<li><a href="Superperfect_number" title="Superperfect number">Superperfect</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Prime_omega_function" title="Prime omega function">Prime omega functions</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Almost_prime" title="Almost prime">Almost prime</a></li>

<li><a href="Semiprime" title="Semiprime">Semiprime</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Euler's_totient_function" title="Euler's totient function">Euler's totient function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li>

<li><a href="Highly_totient_number" title="Highly totient number">Highly totient</a></li>

<li><a href="Noncototient" title="Noncototient">Noncototient</a></li>

<li><a href="Nontotient" title="Nontotient">Nontotient</a></li>

<li><a href="Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li>

<li><a href="Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amicable_numbers" title="Amicable numbers">Amicable</a></li>

<li><a href="Perfect_number" title="Perfect number">Perfect</a></li>

<li><a href="Sociable_numbers" class="mw-redirect" title="Sociable numbers">Sociable</a></li>

<li><a href="Untouchable_number" title="Untouchable number">Untouchable</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Primorial" title="Primorial">Primorial</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Euclid_number" title="Euclid number">Euclid</a></li>

<li><a href="Fortunate_number" title="Fortunate number">Fortunate</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_prime_factor_or_divisor_related_numbers743" style="font-size:114%;margin:0 4em">Other <a href="Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> or <a href="Divisor" title="Divisor">divisor</a> related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Cyclic_number_(group_theory)" title="Cyclic number (group theory)">Cyclic</a></li>

<li><a href="Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li>

<li><a href="Erd%C5%91s%E2%80%93Woods_number" title="Erdős–Woods number">Erdős–Woods</a></li>

<li><a href="Friendly_number" title="Friendly number">Friendly</a></li>

<li><a href="Giuga_number" title="Giuga number">Giuga</a></li>

<li><a href="Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li>

<li><a href="Jordan%E2%80%93P%C3%B3lya_number" title="Jordan–Pólya number">Jordan–Pólya</a></li>

<li><a href="Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael</a></li>

<li><a href="Pronic_number" title="Pronic number">Pronic</a></li>

<li><a href="Regular_number" title="Regular number">Regular</a></li>

<li><a href="Rough_number" title="Rough number">Rough</a></li>

<li><a href="Smooth_number" title="Smooth number">Smooth</a></li>

<li><a href="Sphenic_number" title="Sphenic number">Sphenic</a></li>

<li><a href="St%C3%B8rmer_number" title="Størmer number">Størmer</a></li>

<li><a href="Super-Poulet_number" title="Super-Poulet number">Super-Poulet</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numeral_system-dependent_numbers743" style="font-size:114%;margin:0 4em"><a href="Numeral_system" title="Numeral system">Numeral system</a>-dependent numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Persistence_of_a_number" title="Persistence of a number">Persistence</a>

<ul><li><a href="Additive_persistence" class="mw-redirect" title="Additive persistence">Additive</a></li>

<li><a href="Multiplicative_persistence" class="mw-redirect" title="Multiplicative persistence">Multiplicative</a></li></ul></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Digit_sum" title="Digit sum">Digit sum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Digit_sum" title="Digit sum">Digit sum</a></li>

<li><a href="Digital_root" title="Digital root">Digital root</a></li>

<li><a href="Sum-product_number" title="Sum-product number">Sum-product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit product</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Multiplicative_digital_root" title="Multiplicative digital root">Multiplicative digital root</a></li>

<li><a href="Sum-product_number" title="Sum-product number">Sum-product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coding-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Meertens_number" title="Meertens number">Meertens</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dudeney_number" title="Dudeney number">Dudeney</a></li>

<li><a href="Factorion" title="Factorion">Factorion</a></li>

<li><a href="Kaprekar_number" title="Kaprekar number">Kaprekar</a></li>

<li><a href="Kaprekar's_routine" title="Kaprekar's routine">Kaprekar's constant</a></li>

<li><a href="Keith_number" title="Keith number">Keith</a></li>

<li><a href="Lychrel_number" title="Lychrel number">Lychrel</a></li>

<li><a href="Narcissistic_number" title="Narcissistic number">Narcissistic</a></li>

<li><a href="Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li>

<li><a href="Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a>

<ul><li><a href="Happy_number" title="Happy number">Happy</a></li></ul></li></ul>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Automorphic_number" title="Automorphic number">Automorphic</a>

<ul><li><a href="Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Numerical_digit" title="Numerical digit">Digit</a>-composition related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Palindromic_number" title="Palindromic number">Palindromic</a></li>

<li><a href="Pandigital_number" title="Pandigital number">Pandigital</a></li>

<li><a href="Repdigit" title="Repdigit">Repdigit</a></li>

<li><a href="Repunit" title="Repunit">Repunit</a></li>

<li><a href="Self-descriptive_number" title="Self-descriptive number">Self-descriptive</a></li>

<li><a href="Smarandache%E2%80%93Wellin_number" title="Smarandache–Wellin number">Smarandache–Wellin</a></li>

<li><a href="Undulating_number" title="Undulating number">Undulating</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit-<a href="Permutation" title="Permutation">permutation</a> related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cyclic_number" title="Cyclic number">Cyclic</a></li>

<li><a href="Digit-reassembly_number" title="Digit-reassembly number">Digit-reassembly</a></li>

<li><a href="Parasitic_number" title="Parasitic number">Parasitic</a></li>

<li><a href="Primeval_number" title="Primeval number">Primeval</a></li>

<li><a href="Transposable_integer" title="Transposable integer">Transposable</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisor-related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Equidigital_number" title="Equidigital number">Equidigital</a></li>

<li><a href="Extravagant_number" title="Extravagant number">Extravagant</a></li>

<li><a href="Frugal_number" title="Frugal number">Frugal</a></li>

<li><a href="Harshad_number" title="Harshad number">Harshad</a></li>

<li><a href="Polydivisible_number" title="Polydivisible number">Polydivisible</a></li>

<li><a href="Smith_number" title="Smith number">Smith</a></li>

<li><a href="Vampire_number" title="Vampire number">Vampire</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Friedman_number" title="Friedman number">Friedman</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Binary_numbers743" style="font-size:114%;margin:0 4em"><a href="Binary_number" title="Binary number">Binary numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Odious_number" title="Odious number">Odious</a></li>

<li><a href="Pernicious_number" title="Pernicious number">Pernicious</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve743" style="font-size:114%;margin:0 4em">Generated via a <a href="Sieve_theory" title="Sieve theory">sieve</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Lucky_number" title="Lucky number">Lucky</a></li>

<li><a href="Generation_of_primes" title="Generation of primes">Prime</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related743" style="font-size:114%;margin:0 4em"><a href="Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pancake_sorting" title="Pancake sorting">Pancake number</a></li>

<li><a href="Sorting_number" title="Sorting number">Sorting number</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related743" style="font-size:114%;margin:0 4em"><a href="Natural_language" title="Natural language">Natural language</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Aronson's_sequence" title="Aronson's sequence">Aronson's sequence</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related743" style="font-size:114%;margin:0 4em"><a href="Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Strobogrammatic_number" title="Strobogrammatic number">Strobogrammatic</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2" style="font-weight:bold;"><div>

<ul><li> <a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul>

</div></td></tr></tbody></table></div>

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This article is issued from <a class="external text" title="Last edited on 2025-07-29" href="https://en.wikipedia.org/wiki/?title=Exponentiation&oldid=1303187782">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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