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nth root

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about nth-roots of real and complex numbers. For other uses, see <a href="Root_(disambiguation)#Mathematics" class="mw-disambig" title="Root (disambiguation)">Root (disambiguation) § Mathematics</a>.</div>

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In <a href="Mathematics" title="Mathematics">mathematics</a>, an nth root of a <a href="Number" title="Number">number</a> x is a number r which, when <a href="Exponentiation" title="Exponentiation">raised to the power</a> of n, yields x: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{n}=\underbrace {r\times r\times \dotsb \times r} _{n{\text{ factors}}}=x.}">

<msup>

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<mtext> factors</mtext>

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

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The <a href="Positive_integer" class="mw-redirect" title="Positive integer">positive integer</a> n is called the index or degree, and the number x of which the root is taken is the radicand. A root of degree 2 is called a <a href="Square_root" title="Square root">square root</a> and a root of degree 3, a <a href="Cube_root" title="Cube root">cube root</a>. Roots of higher degree are referred by using <a href="Ordinal_numeral" title="Ordinal numeral">ordinal numbers</a>, as in fourth root, twentieth root, etc. The computation of an nth root is a root extraction.

For example, 3 is a square root of 9, since 32 = 9, and −3 is also a square root of 9, since (−3)2 = 9.

The nth root of x is written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}}">

<mroot>

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

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</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"> using the <a href="Radical_symbol" title="Radical symbol">radical symbol</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\phantom {x}}}}">

<msqrt>

</mphantom>

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</msqrt>

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

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</math><img src="./b8f3297b4c8819ba4e37691292362bbaf312906a.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 {\phantom {x}}}}" loading="lazy">. The square root is usually written as ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x}}}">

<msqrt>

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

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</math><img src="./d62b24be305beff66cba9bfbcc01a362ba390f44.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 {x}}}" loading="lazy">⁠, with the degree omitted. Taking the nth root of a number, for fixed ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</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 the <a href="Inverse_function#Squaring_and_square_root_functions" title="Inverse function">inverse</a> of raising a number to the nth power,<a href="#cite_note-1">[1]</a> and can be written as a <a href="Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fractional</a> exponent:

<mroot>

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</mroot>

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<msup>

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

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For a positive real number x, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x}}}">

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

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</math><img src="./d62b24be305beff66cba9bfbcc01a362ba390f44.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 {x}}}" loading="lazy"> denotes the positive square root of x and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}}">

<mroot>

</mrow>

</mroot>

</mrow>

</mstyle>

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

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</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 positive real nth root. A negative real number −x has no real-valued square roots, but when x is treated as a complex number it has two <a href="Imaginary_number" title="Imaginary number">imaginary</a> square roots, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +i{\sqrt {x}}}">

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

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</math><img src="./9481b44f7e22368b5b8c2a284f3c5a80e3e954cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.876ex; height:3.009ex;" alt="{\displaystyle +i{\sqrt {x}}}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i{\sqrt {x}}}">

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

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</math><img src="./acb49ee97737a5c4a425b013b2117d0b04891dae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.876ex; height:3.009ex;" alt="{\displaystyle -i{\sqrt {x}}}" loading="lazy">⁠, where i is the <a href="Imaginary_unit" title="Imaginary unit">imaginary unit</a>.

In general, any non-zero <a href="Complex_number" title="Complex number">complex number</a> has n distinct complex-valued nth roots, equally distributed around a complex circle of constant <a href="Absolute_value#Complex_numbers" title="Absolute value">absolute value</a>. (The nth root of 0 is zero with <a href="Multiple_root" class="mw-redirect" title="Multiple root">multiplicity</a> n, and this circle degenerates to a point.) Extracting the nth roots of a complex number x can thus be taken to be a <a href="Multivalued_function" title="Multivalued function">multivalued function</a>. By convention the <a href="Principal_value" title="Principal value">principal value</a> of this function, called the principal root and denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}}">

<mroot>

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

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</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">⁠, is taken to be the nth root with the greatest real part and in the special case when x is a negative real number, the one with a positive <a href="Imaginary_part" class="mw-redirect" title="Imaginary part">imaginary part</a>. The principal root of a positive real number is thus also a positive real number. As a <a href="Function_(mathematics)" title="Function (mathematics)">function</a>, the principal root is <a href="Continuous_function" title="Continuous function">continuous</a> in the whole <a href="Complex_plane" title="Complex plane">complex plane</a>, except along the negative real axis.

An unresolved root, especially one using the radical symbol, is sometimes referred to as a surd<a href="#cite_note-2">[2]</a> or a radical.<a href="#cite_note-silver-3">[3]</a> Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called a radical expression, and if it contains no <a href="Transcendental_functions" class="mw-redirect" title="Transcendental functions">transcendental functions</a> or <a href="Transcendental_numbers" class="mw-redirect" title="Transcendental numbers">transcendental numbers</a> it is called an <a href="Algebraic_expression" title="Algebraic expression">algebraic expression</a>.

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<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;">

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<mtext>summand</mtext>

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<mtext>summand</mtext>

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</mtr>

<mtr>

<mtd>

<mtext>addend</mtext>

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<mtext>addend</mtext>

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<mtr>

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<mtext>augend</mtext>

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<mtext>addend</mtext>

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<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>

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</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;">

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

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</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;">

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<mtr>

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<mtext>minuend</mtext>

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<mtext>subtrahend</mtext>

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<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"><a href="Exponentiation" title="Exponentiation">Exponentiation</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>

<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"> (√)</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>

</tr><tr><td class="sidebar-navbar"></td></tr></tbody></table>

Roots are used for determining the <a href="Radius_of_convergence" title="Radius of convergence">radius of convergence</a> of a <a href="Power_series" title="Power series">power series</a> with the <a href="Root_test" title="Root test">root test</a>. The nth roots of 1 are called <a href="Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a> and play a fundamental role in various areas of mathematics, such as <a href="Number_theory" title="Number theory">number theory</a>, <a href="Theory_of_equations" title="Theory of equations">theory of equations</a>, and <a href="Fourier_transform" title="Fourier transform">Fourier transform</a>.

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

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Square_root#History" title="Square root">Square root § History</a>, and <a href="Cube_root#History" title="Cube root">Cube root § History</a></div>

An archaic term for the operation of taking nth roots is radication.<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition_and_notation">Definition and notation</h2></div>

An nth root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose nth power is x:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math>

Every positive <a href="Real_number" title="Real number">real number</a> x has a single positive nth root, called the <a href="Principal_value" title="Principal value">principal nth root</a>, which is written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}}">

<mroot>

</mrow>

</mroot>

</mrow>

</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">. For n equal to 2 this is called the principal square root and the n is omitted. The nth root can also be represented using <a href="Exponentiation" title="Exponentiation">exponentiation</a> as x1/n.

For even values of n, positive numbers also have a negative nth root, while negative numbers do not have a real nth root. For odd values of n, every negative number x has a real negative nth root. For example, −2 has a real 5th root, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{5}]{-2}}=-1.148698354\ldots }">

<mroot>

<mrow>

</mrow>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt[{5}]{-2}}=-1.148698354\ldots }</annotation>

</semantics>

</math><img src="./98d22c0a8f77736a738e9566bd1ebd1b46438ffb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.195ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{5}]{-2}}=-1.148698354\ldots }" loading="lazy"> but −2 does not have any real 6th roots.

Every non-zero number x, real or <a href="Complex_number" title="Complex number">complex</a>, has n different complex number nth roots. (In the case x is real, this count includes any real nth roots.) The only complex root of 0 is 0.

The nth roots of almost all numbers (all integers except the nth powers, and all rationals except the quotients of two nth powers) are <a href="Irrational_number" title="Irrational number">irrational</a>. For example,

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}=1.414213562\ldots }</annotation>

</semantics>

</math>

All nth roots of rational numbers are <a href="Algebraic_number" title="Algebraic number">algebraic numbers</a>, and all nth roots of integers are <a href="Algebraic_integer" title="Algebraic integer">algebraic integers</a>.

The term "surd" traces back to <a href="Al-Khwarizmi" title="Al-Khwarizmi">Al-Khwarizmi</a> (<abbr title="circa">c.</abbr> 825), who referred to rational and irrational numbers as "audible" and "inaudible", respectively. This later led to the Arabic word أصم (asamm, meaning "deaf" or "dumb") for "irrational number" being translated into Latin as surdus (meaning "deaf" or "mute"). <a href="Gerard_of_Cremona" title="Gerard of Cremona">Gerard of Cremona</a> (<abbr title="circa">c.</abbr> 1150), <a href="Fibonacci" title="Fibonacci">Fibonacci</a> (1202), and then <a href="Robert_Recorde" title="Robert Recorde">Robert Recorde</a> (1551) all used the term to refer to "unresolved irrational roots", that is, expressions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{r}}}">

<mroot>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./10eb7386bd8efe4c5b5beafe05848fbd923e1413.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.985ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{r}}}" loading="lazy">, in which <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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> are integer numerals and the whole expression denotes an irrational number.<a href="#cite_note-6">[6]</a> Irrational numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\sqrt {a}},}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./502ee72ef0c6df79861cb698b43ec75c8580a50c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.621ex; height:3.009ex;" alt="{\displaystyle \pm {\sqrt {a}},}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> is rational, are called "pure quadratic surds"; irrational numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\pm {\sqrt {b}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4489180f5f56df7467de36d3af2903961317b93f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.004ex; height:3.009ex;" alt="{\displaystyle a\pm {\sqrt {b}}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</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"> are rational, are called <a href="Quadratic_irrational_number" title="Quadratic irrational number">mixed quadratic surds</a>.<a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading3"><h3 id="Square_roots">Square roots</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Square_root" title="Square root">Square root</a></div>

A square root of a number x is a number r which, when <a href="Square_(algebra)" title="Square (algebra)">squared</a>, becomes x:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math>

Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as the principal square root, and is denoted with a radical sign:

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {25}}=5.}</annotation>

</semantics>

</math>

Since the square of every real number is nonnegative, negative numbers do not have real square roots. However, for every negative real number there are two <a href="Imaginary_number" title="Imaginary number">imaginary</a> square roots. For example, the square roots of −25 are 5i and −5i, where <a href="Imaginary_unit" title="Imaginary unit">i</a> represents a number whose square is −1.

<div class="mw-heading mw-heading3"><h3 id="Cube_roots">Cube roots</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Cube_root" title="Cube root">Cube root</a></div>

A cube root of a number x is a number r whose <a href="Cube_(algebra)" title="Cube (algebra)">cube</a> is x:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math>

Every real number x has exactly one real cube root, written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{x}}}">

<mroot>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{x}}}</annotation>

</semantics>

</math><img src="./9a55f866116e7a86823816615dd98fcccde75473.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[{3}]{x}}}" loading="lazy">. For example,

<mtr>

<mtd>

<mroot>

</mrow>

</mroot>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mroot>

<mrow>

</mrow>

</mrow>

</mroot>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\sqrt[{3}]{8}}&=2\\{\sqrt[{3}]{-8}}&=-2.\end{aligned}}}</annotation>

</semantics>

</math>

Every real number has two additional <a href="Complex_number" title="Complex number">complex</a> cube roots.

<div class="mw-heading mw-heading2"><h2 id="Identities_and_properties">Identities and properties</h2></div>

Expressing the degree of an nth root in its exponent form, as in <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">, makes it easier to manipulate powers and roots. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> is a <a href="Non-negative_number" class="mw-redirect" title="Non-negative number">non-negative real number</a>,

<mroot>

<msup>

</mrow>

</msup>

</mrow>

</mroot>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mroot>

</mrow>

</mroot>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{a^{m}}}=(a^{m})^{1/n}=a^{m/n}=(a^{1/n})^{m}=({\sqrt[{n}]{a}})^{m}.}</annotation>

</semantics>

</math>

Every non-negative number has exactly one non-negative real nth root, and so the rules for operations with surds involving non-negative radicands <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</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"> are straightforward within the real numbers:

<mtr>

<mtd>

<mroot>

<mrow>

</mrow>

</mrow>

</mroot>

</mrow>

</mtd>

<mtd>

<mroot>

</mrow>

</mroot>

</mrow>

<mroot>

</mrow>

</mroot>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mroot>

<mfrac>

</mfrac>

</mrow>

</mroot>

</mrow>

</mtd>

<mtd>

<mfrac>

<mroot>

</mrow>

</mroot>

<mroot>

</mrow>

</mroot>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\sqrt[{n}]{ab}}&={\sqrt[{n}]{a}}{\sqrt[{n}]{b}}\\{\sqrt[{n}]{\frac {a}{b}}}&={\frac {\sqrt[{n}]{a}}{\sqrt[{n}]{b}}}\end{aligned}}}</annotation>

</semantics>

</math>

Subtleties can occur when taking the nth roots of negative or <a href="Complex_number" title="Complex number">complex numbers</a>. For instance:

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {-1}}\times {\sqrt {-1}}\neq {\sqrt {-1\times -1}}=1,\quad }</annotation>

</semantics>

</math>

but, rather,

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \quad {\sqrt {-1}}\times {\sqrt {-1}}=i\times i=i^{2}=-1.}</annotation>

</semantics>

</math>

Since the rule <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{a}}\times {\sqrt[{n}]{b}}={\sqrt[{n}]{ab}}}">

<mroot>

</mrow>

</mroot>

</mrow>

<mroot>

</mrow>

</mroot>

</mrow>

<mroot>

<mrow>

</mrow>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7d0a49ffbfd95598ffe89e29489a3d475de5fb58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.201ex; height:3.343ex;" alt="{\displaystyle {\sqrt[{n}]{a}}\times {\sqrt[{n}]{b}}={\sqrt[{n}]{ab}}}" loading="lazy"> strictly holds for non-negative real radicands only, its application leads to the inequality in the first step above.

<div class="mw-heading mw-heading2"><h2 id="Simplified_form_of_a_radical_expression">Simplified form of a radical expression</h2></div>

A <a href="Nested_radical" title="Nested radical">non-nested radical expression</a> is said to be in simplified form if no factor of the radicand can be written as a power greater than or equal to the index; there are no fractions inside the radical sign; and there are no radicals in the denominator.<a href="#cite_note-8">[8]</a>

For example, to write the radical expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\sqrt {32/5}}}">

<msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\sqrt {32/5}}}</annotation>

</semantics>

</math><img src="./97aac1b44ece0660ca49b97fe73fbe93487c1c89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.974ex; height:3.343ex;" alt="{\displaystyle \textstyle {\sqrt {32/5}}}" loading="lazy"> in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it:

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

<msqrt>

<mfrac>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\frac {32}{5}}}={\sqrt {\frac {16\cdot 2}{5}}}={\sqrt {16}}\cdot {\sqrt {\frac {2}{5}}}=4{\sqrt {\frac {2}{5}}}}</annotation>

</semantics>

</math>

Next, there is a fraction under the radical sign, which we change as follows:

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4{\sqrt {\frac {2}{5}}}={\frac {4{\sqrt {2}}}{\sqrt {5}}}}</annotation>

</semantics>

</math>

Finally, we remove the radical from the denominator as follows:

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mfrac>

<msqrt>

</msqrt>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {4{\sqrt {2}}}{\sqrt {5}}}={\frac {4{\sqrt {2}}}{\sqrt {5}}}\cdot {\frac {\sqrt {5}}{\sqrt {5}}}={\frac {4{\sqrt {10}}}{5}}={\frac {4}{5}}{\sqrt {10}}}</annotation>

</semantics>

</math>

When there is a denominator involving surds it is always possible to find a factor to multiply both numerator and denominator by to simplify the expression.<a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a> For instance using the <a href="Factorization#Sum/difference_of_two_cubes" title="Factorization">factorization of the sum of two cubes</a>:

<mfrac>

<mrow>

<mroot>

</mrow>

</mroot>

</mrow>

<mroot>

</mrow>

</mroot>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mroot>

<msup>

</mrow>

</msup>

</mrow>

</mroot>

</mrow>

<mroot>

<mrow>

</mrow>

</mrow>

</mroot>

</mrow>

<mroot>

<msup>

</mrow>

</msup>

</mrow>

</mroot>

</mrow>

<mrow>

<mrow>

<mroot>

</mrow>

</mroot>

</mrow>

<mroot>

</mrow>

</mroot>

</mrow>

</mrow>

<mrow>

<mrow>

<mroot>

<msup>

</mrow>

</msup>

</mrow>

</mroot>

</mrow>

<mroot>

<mrow>

</mrow>

</mrow>

</mroot>

</mrow>

<mroot>

<msup>

</mrow>

</msup>

</mrow>

</mroot>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mroot>

<msup>

</mrow>

</msup>

</mrow>

</mroot>

</mrow>

<mroot>

<mrow>

</mrow>

</mrow>

</mroot>

</mrow>

<mroot>

<msup>

</mrow>

</msup>

</mrow>

</mroot>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{{\sqrt[{3}]{a}}+{\sqrt[{3}]{b}}}}={\frac {{\sqrt[{3}]{a^{2}}}-{\sqrt[{3}]{ab}}+{\sqrt[{3}]{b^{2}}}}{\left({\sqrt[{3}]{a}}+{\sqrt[{3}]{b}}\right)\left({\sqrt[{3}]{a^{2}}}-{\sqrt[{3}]{ab}}+{\sqrt[{3}]{b^{2}}}\right)}}={\frac {{\sqrt[{3}]{a^{2}}}-{\sqrt[{3}]{ab}}+{\sqrt[{3}]{b^{2}}}}{a+b}}.}</annotation>

</semantics>

</math>

Simplifying radical expressions involving <a href="Nested_radical" title="Nested radical">nested radicals</a> can be quite difficult. In particular, denesting is not always possible, and when possible, it may involve advanced <a href="Galois_theory" title="Galois theory">Galois theory</a>. Moreover, when complete denesting is impossible, there is no general <a href="Canonical_form" title="Canonical form">canonical form</a> such that the equality of two numbers can be tested by simply looking at their canonical expressions.

For example, it is not obvious that

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {3+2{\sqrt {2}}}}=1+{\sqrt {2}}.}</annotation>

</semantics>

</math>

The above can be derived through:

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msqrt>

</msqrt>

</mrow>

<msup>

<msqrt>

</msqrt>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {3+2{\sqrt {2}}}}={\sqrt {1+2{\sqrt {2}}+2}}={\sqrt {1^{2}+2{\sqrt {2}}+{\sqrt {2}}^{2}}}={\sqrt {\left(1+{\sqrt {2}}\right)^{2}}}=1+{\sqrt {2}}}</annotation>

</semantics>

</math>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=p/q}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1696e05a9fb034877500b9f7a2ff6c3a0c2564a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.549ex; height:2.843ex;" alt="{\displaystyle r=p/q}" loading="lazy">, with p and q coprime and positive integers. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{r}}={\sqrt[{n}]{p}}/{\sqrt[{n}]{q}}}">

<mroot>

</mrow>

</mroot>

</mrow>

<mroot>

</mrow>

</mroot>

</mrow>

</mrow>

<mroot>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./32f9eceaf6392a34ca84e490204f6eef56b4a7be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.356ex; height:3.176ex;" alt="{\displaystyle {\sqrt[{n}]{r}}={\sqrt[{n}]{p}}/{\sqrt[{n}]{q}}}" loading="lazy"> is rational if and only if both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{p}}}">

<mroot>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dac134c13bde44d42060499220adf6949490f40e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.105ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{p}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{q}}}">

<mroot>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3dfc3fcfbe3811c3e980414f3a6c90ca7c286ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.005ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{q}}}" loading="lazy"> are integers, which means that both p and q are nth powers of some integer.

<div class="mw-heading mw-heading2"><h2 id="Infinite_series">Infinite series</h2></div>

The radical or root may be represented by the <a href="Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a>:

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1+x)^{s/t}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}\prod _{k=0}^{n-1}\left({\frac {s}{t}}-k\right)}</annotation>

</semantics>

</math>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|<1}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2f9498d60b2319a4ae7c5607794b537c559a976d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle |x|<1}" loading="lazy">. This expression can be derived from the <a href="Binomial_series" title="Binomial series">binomial series</a>.

<div class="mw-heading mw-heading2"><h2 id="Computing_principal_roots">Computing principal roots</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Using_Newton's_method">Using Newton's method</h3></div>

The nth root of a number A can be computed with <a href="Newton's_method" title="Newton's method">Newton's method</a>, which starts with an initial guess x0 and then iterates using the <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{k+1}=x_{k}-{\frac {x_{k}^{n}-A}{nx_{k}^{n-1}}}}</annotation>

</semantics>

</math>

until the desired precision is reached. For computational efficiency, the recurrence relation is commonly rewritten

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mrow>

</mrow>

</msubsup>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math>

This allows to have only one <a href="Exponentiation" title="Exponentiation">exponentiation</a>, and to compute once for all the first factor of each term.

For example, to find the fifth root of 34, we plug in n = 5, A = 34 and x0 = 2 (initial guess). The first 5 iterations are, approximately:

/* start https://en.wikipedia.org/ */

.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}

/* end https://en.wikipedia.org/ */

</style><div class="block-indent">x0 = 2</div>

(All correct digits shown.)

The approximation x4 is accurate to 25 decimal places and x5 is good for 51.

Newton's method can be modified to produce various <a href="Generalized_continued_fraction" class="mw-redirect" title="Generalized continued fraction">generalized continued fractions</a> for the nth root. For example,

<mroot>

</mrow>

</mroot>

</mrow>

<mroot>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mroot>

</mrow>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{z}}={\sqrt[{n}]{x^{n}+y}}=x+{\cfrac {y}{nx^{n-1}+{\cfrac {(n-1)y}{2x+{\cfrac {(n+1)y}{3nx^{n-1}+{\cfrac {(2n-1)y}{2x+{\cfrac {(2n+1)y}{5nx^{n-1}+{\cfrac {(3n-1)y}{2x+\ddots }}}}}}}}}}}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Digit-by-digit_calculation_of_principal_roots_of_decimal_(base_10)_numbers">Digit-by-digit calculation of principal roots of decimal (base 10) numbers</h3></div>

Building on the <a href="Methods_of_computing_square_roots" class="mw-redirect" title="Methods of computing square roots">digit-by-digit calculation of a square root</a>, it can be seen that the formula used there, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(20p+x)\leq c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(20p+x)\leq c}</annotation>

</semantics>

</math><img src="./4cfe39e34c2c6ee28294d1ff2195e3dfbca33cac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.909ex; height:2.843ex;" alt="{\displaystyle x(20p+x)\leq c}" loading="lazy">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+20xp\leq c}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}+20xp\leq c}</annotation>

</semantics>

</math><img src="./cd3b37b91f0f08498167cb382c372e2944e55892.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.154ex; height:3.009ex;" alt="{\displaystyle x^{2}+20xp\leq c}" loading="lazy">, follows a pattern involving Pascal's triangle. For the nth root of a number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n,i)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d2315f4ec75ea3037b5cf5c71ab7c4f88004c12e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.786ex; height:2.843ex;" alt="{\displaystyle P(n,i)}" loading="lazy"> is defined as the value of element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</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"> in row <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"> of Pascal's Triangle such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(4,1)=4}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(4,1)=4}</annotation>

</semantics>

</math><img src="./b841cdb2c861763e0e9f09d28d568adbecbbab0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.174ex; height:2.843ex;" alt="{\displaystyle P(4,1)=4}" loading="lazy">, we can rewrite the expression as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{n-1}10^{i}P(n,i)p^{i}x^{n-i}}">

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n-1}10^{i}P(n,i)p^{i}x^{n-i}}</annotation>

</semantics>

</math><img src="./000d6d39fce76fabc686702ebea16892d10b5028.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.016ex; height:7.343ex;" alt="{\displaystyle \sum _{i=0}^{n-1}10^{i}P(n,i)p^{i}x^{n-i}}" loading="lazy">. For convenience, call the result of this expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">. Using this more general expression, any positive principal root can be computed, digit-by-digit, as follows.

Write the original number in decimal form. The numbers are written similar to the <a href="Long_division" title="Long division">long division</a> algorithm, and, as in long division, the root will be written on the line above. Now separate the digits into groups of digits equating to the root being taken, starting from the decimal point and going both left and right. The decimal point of the root will be above the decimal point of the radicand. One digit of the root will appear above each group of digits of the original number.

Beginning with the left-most group of digits, do the following procedure for each group:

<ol><li>Starting on the left, bring down the most significant (leftmost) group of digits not yet used (if all the digits have been used, write "0" the number of times required to make a group) and write them to the right of the remainder from the previous step (on the first step, there will be no remainder). In other words, multiply the remainder by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 10^{n}}</annotation>

</semantics>

</math><img src="./eeff06a7c9ad9455cb809047cfc97a92c51e1bf7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.343ex;" alt="{\displaystyle 10^{n}}" loading="lazy"> and add the digits from the next group. This will be the current value c.</li>

<li>Find p and x, as follows:

</mstyle>

</mrow>

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

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> be the part of the root found so far, ignoring any decimal point. (For the first step, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b3e6ac10fa45fb984d886065f959a6bdd467b5e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{0}=1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./478100bc5766c6af537439ef9309f9ddf2f9a6ff.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 0^{0}=1}" loading="lazy">).</li>

<li>Determine the greatest digit <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"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\leq c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./602c121cbacae4c9b48807a3baf4dece313dca2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.261ex; height:2.343ex;" alt="{\displaystyle y\leq c}" loading="lazy">.</li>

<li>Place the digit <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"> as the next digit of the root, i.e., above the group of digits you just brought down. Thus the next p will be the old p times 10 plus x.</li></ul></li>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> to form a new remainder.</li>

<li>If the remainder is zero and there are no more digits to bring down, then the algorithm has terminated. Otherwise go back to step 1 for another iteration.</li></ol>

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

Find the square root of 152.2756.

<pre> 1 2. 3 4

/

\/ 01 52.27 56 (Results) (Explanations)

01 x = 1 100·1·00·12 + 101·2·01·11 ≤ 1 < 100·1·00·22 + 101·2·01·21

01 y = 1 y = 100·1·00·12 + 101·2·01·11 = 1 + 0 = 1

00 52 x = 2 100·1·10·22 + 101·2·11·21 ≤ 52 < 100·1·10·32 + 101·2·11·31

00 44 y = 44 y = 100·1·10·22 + 101·2·11·21 = 4 + 40 = 44

08 27 x = 3 100·1·120·32 + 101·2·121·31 ≤ 827 < 100·1·120·42 + 101·2·121·41

07 29 y = 729 y = 100·1·120·32 + 101·2·121·31 = 9 + 720 = 729

98 56 x = 4 100·1·1230·42 + 101·2·1231·41 ≤ 9856 < 100·1·1230·52 + 101·2·1231·51

98 56 y = 9856 y = 100·1·1230·42 + 101·2·1231·41 = 16 + 9840 = 9856

00 00

</pre>

Algorithm terminates: Answer is 12.34

Find the cube root of 4192 truncated to the nearest thousandth.

<pre> 1 6. 1 2 4

3 /

\/ 004 192.000 000 000 (Results) (Explanations)

004 x = 1 100·1·00·13 + 101·3·01·12 + 102·3·02·11 ≤ 4 < 100·1·00·23 + 101·3·01·22 + 102·3·02·21

001 y = 1 y = 100·1·00·13 + 101·3·01·12 + 102·3·02·11 = 1 + 0 + 0 = 1

003 192 x = 6 100·1·10·63 + 101·3·11·62 + 102·3·12·61 ≤ 3192 < 100·1·10·73 + 101·3·11·72 + 102·3·12·71

003 096 y = 3096 y = 100·1·10·63 + 101·3·11·62 + 102·3·12·61 = 216 + 1,080 + 1,800 = 3,096

096 000 x = 1 100·1·160·13 + 101·3·161·12 + 102·3·162·11 ≤ 96000 < 100·1·160·23 + 101·3·161·22 + 102·3·162·21

077 281 y = 77281 y = 100·1·160·13 + 101·3·161·12 + 102·3·162·11 = 1 + 480 + 76,800 = 77,281

018 719 000 x = 2 100·1·1610·23 + 101·3·1611·22 + 102·3·1612·21 ≤ 18719000 < 100·1·1610·33 + 101·3·1611·32 + 102·3·1612·31

015 571 928 y = 15571928 y = 100·1·1610·23 + 101·3·1611·22 + 102·3·1612·21 = 8 + 19,320 + 15,552,600 = 15,571,928

003 147 072 000 x = 4 100·1·16120·43 + 101·3·16121·42 + 102·3·16122·41 ≤ 3147072000 < 100·1·16120·53 + 101·3·16121·52 + 102·3·16122·51

</pre>

The desired precision is achieved. The cube root of 4192 is 16.124...

<div class="mw-heading mw-heading3"><h3 id="Logarithmic_calculation">Logarithmic calculation</h3></div>

The principal nth root of a positive number can be computed using <a href="Logarithm" title="Logarithm">logarithms</a>. Starting from the equation that defines r as an nth root of x, namely <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{n}=x,}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c22cbcdc8f114887cf0ceeb723bb9624c6f132e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.342ex; height:2.676ex;" alt="{\displaystyle r^{n}=x,}" loading="lazy"> with x positive and therefore its principal root r also positive, one takes logarithms of both sides (any <a href="Logarithm#Particular_bases" title="Logarithm">base of the logarithm</a> will do) to obtain

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mtext>hence</mtext>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\log _{b}r=\log _{b}x\quad \quad {\text{hence}}\quad \quad \log _{b}r={\frac {\log _{b}x}{n}}.}</annotation>

</semantics>

</math>

The root r is recovered from this by taking the <a href="Antilog" class="mw-redirect" title="Antilog">antilog</a>:

<msup>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math>

(Note: That formula shows b raised to the power of the result of the division, not b multiplied by the result of the division.)

For the case in which x is negative and n is odd, there is one real root r which is also negative. This can be found by first multiplying both sides of the defining equation by −1 to obtain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |r|^{n}=|x|,}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./070955f8d31c41a72920225b4da754731e240c4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.929ex; height:3.009ex;" alt="{\displaystyle |r|^{n}=|x|,}" loading="lazy"> then proceeding as before to find |r|, and using r = −|r|.

<div class="mw-heading mw-heading2"><h2 id="Geometric_constructibility">Geometric constructibility</h2></div>

The <a href="Ancient_Greek_mathematicians" class="mw-redirect" title="Ancient Greek mathematicians">ancient Greek mathematicians</a> knew how to <a href="Compass-and-straightedge_construction" class="mw-redirect" title="Compass-and-straightedge construction">use compass and straightedge</a> to construct a length equal to the square root of a given length, when an auxiliary line of unit length is given. In 1837 <a href="Pierre_Wantzel" title="Pierre Wantzel">Pierre Wantzel</a> proved that an nth root of a given length cannot be constructed if n is not a power of 2.<a href="#cite_note-11">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Complex_roots">Complex roots</h2></div>

Every <a href="Complex_number" title="Complex number">complex number</a> other than 0 has n different nth roots.

<div class="mw-heading mw-heading3"><h3 id="Square_roots_2">Square roots</h3></div>

The two square roots of a complex number are always negatives of each other. For example, the square roots of −4 are 2i and −2i, and the square roots of i are

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mstyle>

</mrow>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {2}}}(1+i)\quad {\text{and}}\quad -{\tfrac {1}{\sqrt {2}}}(1+i).}</annotation>

</semantics>

</math>

If we express a complex number in <a href="Polar_form" class="mw-redirect" title="Polar form">polar form</a>, then the square root can be obtained by taking the square root of the radius and halving the angle:

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {re^{i\theta }}}=\pm {\sqrt {r}}\cdot e^{i\theta /2}.}</annotation>

</semantics>

</math>

A principal root of a complex number may be chosen in various ways, for example

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {re^{i\theta }}}={\sqrt {r}}\cdot e^{i\theta /2}}</annotation>

</semantics>

</math>

which introduces a <a href="Branch_cut" class="mw-redirect" title="Branch cut">branch cut</a> in the <a href="Complex_plane" title="Complex plane">complex plane</a> along the <a href="Positive_real_axis" class="mw-redirect" title="Positive real axis">positive real axis</a> with the condition 0 ≤ θ < 2π, or along the negative real axis with −π < θ ≤ π.

Using the first(last) branch cut the principal square root <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\sqrt {z}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4822ea5fba80988907cf54af2c4c5e51ceffadd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {\sqrt {z}}}" loading="lazy"> maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle z}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ea1aac04b6044a6bc2aa0ca36b4580a19019d46e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.769ex; height:1.343ex;" alt="{\displaystyle \scriptstyle z}" loading="lazy"> to the half plane with non-negative imaginary(real) part. The last branch cut is presupposed in mathematical software like <a href="Matlab" class="mw-redirect" title="Matlab">Matlab</a> or <a href="Scilab" title="Scilab">Scilab</a>.

<div class="mw-heading mw-heading3"><h3 id="Roots_of_unity">Roots of unity</h3></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 number 1 has n different nth roots in the complex plane, namely

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math>

where

<msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =e^{\frac {2\pi i}{n}}=\cos \left({\frac {2\pi }{n}}\right)+i\sin \left({\frac {2\pi }{n}}\right).}</annotation>

</semantics>

</math>

These roots are evenly spaced around the <a href="Unit_circle" title="Unit circle">unit circle</a> in the complex plane, at angles which are multiples of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi /n}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3deb2c1b3bc6edccdf1be853df40c7d21b0fc57b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.052ex; height:2.843ex;" alt="{\displaystyle 2\pi /n}" loading="lazy">. For example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

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

</semantics>

</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">, −1, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./91fddb9f89a520937db3a8821575068cdcc76f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.611ex; height:2.343ex;" alt="{\displaystyle -i}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="nth_roots">nth roots</h3></div>

Every complex number has n different nth roots in the complex plane. These are

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math>

where η is a single nth root, and 1, ω, ω2, ... ωn−1 are the nth roots of unity. For example, the four different fourth roots of 2 are

<mroot>

</mrow>

</mroot>

</mrow>

<mroot>

</mrow>

</mroot>

</mrow>

<mroot>

</mrow>

</mroot>

</mrow>

</mrow>

<mroot>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt[{4}]{2}},\quad i{\sqrt[{4}]{2}},\quad -{\sqrt[{4}]{2}},\quad {\text{and}}\quad -i{\sqrt[{4}]{2}}.}</annotation>

</semantics>

</math>

In <a href="Polar_form" class="mw-redirect" title="Polar form">polar form</a>, a single nth root may be found by the formula

<mroot>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mroot>

</mrow>

<mroot>

</mrow>

</mroot>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{re^{i\theta }}}={\sqrt[{n}]{r}}\cdot e^{i\theta /n}.}</annotation>

</semantics>

</math>

Here r is the magnitude (the modulus, also called the <a href="Absolute_value" title="Absolute value">absolute value</a>) of the number whose root is to be taken; if the number can be written as a+bi then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\sqrt {a^{2}+b^{2}}}}">

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c06427751d7f71ba70ddfae47fb47e6386324ae6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.647ex; height:3.509ex;" alt="{\displaystyle r={\sqrt {a^{2}+b^{2}}}}" loading="lazy">. Also, <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 angle formed as one pivots on the origin counterclockwise from the positive horizontal axis to a ray going from the origin to the number; it has the properties that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta =a/r,}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3b521e513e1d910ee3d8771ae7f8115c571bbca8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.775ex; height:2.843ex;" alt="{\displaystyle \cos \theta =a/r,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \theta =b/r,}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ee2942e6cbe7ef79f8cbc5851f1c8960528102dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.287ex; height:2.843ex;" alt="{\displaystyle \sin \theta =b/r,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \theta =b/a.}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./99629f56d9f2784dac6f3c31c4b782b772e906d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.972ex; height:2.843ex;" alt="{\displaystyle \tan \theta =b/a.}" loading="lazy">

Thus finding nth roots in the complex plane can be segmented into two steps. First, the magnitude of all the nth roots is the nth root of the magnitude of the original number. Second, the angle between the positive horizontal axis and a ray from the origin to one of the nth roots is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta /n}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./20905d7e366eec7ab5418bec2cfc043584f32369.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.648ex; height:2.843ex;" alt="{\displaystyle \theta /n}" loading="lazy">, where <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 angle defined in the same way for the number whose root is being taken. Furthermore, all n of the nth roots are at equally spaced angles from each other.

If n is even, a complex number's nth roots, of which there are an even number, come in <a href="Additive_inverse" title="Additive inverse">additive inverse</a> pairs, so that if a number r1 is one of the nth roots then r2 = −r1 is another. This is because raising the latter's coefficient −1 to the nth power for even n yields 1: that is, (−r1)n = (−1)n × r1n = r1n.

As with square roots, the formula above does not define a <a href="Continuous_function" title="Continuous function">continuous function</a> over the entire complex plane, but instead has a <a href="Branch_cut" class="mw-redirect" title="Branch cut">branch cut</a> at points where θ / n is discontinuous.

<div class="mw-heading mw-heading2"><h2 id="Solving_polynomials">Solving polynomials</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Root-finding_algorithms" class="mw-redirect" title="Root-finding algorithms">Root-finding algorithms</a></div>

It was once <a href="Conjecture" title="Conjecture">conjectured</a> that all <a href="Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equations</a> could be <a href="Algebraic_solution" class="mw-redirect" title="Algebraic solution">solved algebraically</a> (that is, that all roots of a <a href="Polynomial" title="Polynomial">polynomial</a> could be expressed in terms of a finite number of radicals and <a href="Elementary_arithmetic" title="Elementary arithmetic">elementary operations</a>). However, while this is true for third degree polynomials (<a href="Cubic_function" title="Cubic function">cubics</a>) and fourth degree polynomials (<a href="Quartic_function" title="Quartic function">quartics</a>), the <a href="Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a> (1824) shows that this is not true in general when the degree is 5 or greater. For example, the solutions of the equation

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{5}=x+1}</annotation>

</semantics>

</math>

cannot be expressed in terms of radicals. (cf. <a href="Quintic_equation" class="mw-redirect" title="Quintic equation">quintic equation</a>)

<div class="mw-heading mw-heading2"><h2 id="Proof_of_irrationality_for_non-perfect_nth_power_x">Proof of irrationality for non-perfect nth power x</h2></div>

Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}}">

<mroot>

</mrow>

</mroot>

</mrow>

</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"> is rational. That is, it can be reduced to a fraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9fbb66e57f89debc3cde3213de12228971148a93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.066ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}}" loading="lazy">, where a and b are integers without a common factor.

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

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6a17612d66adec36e70a10cbe6afc1bea7669b42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.713ex; height:5.343ex;" alt="{\displaystyle x={\frac {a^{n}}{b^{n}}}}" loading="lazy">.

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

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2687184a7698e75db65a25bea7afd207bff3d03b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.448ex; height:2.343ex;" alt="{\displaystyle a^{n}}" loading="lazy">and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</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">must share a common factor if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\neq 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./32a55c3a4b8eca743088b70c7c4d63c773f43c2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.258ex; height:2.676ex;" alt="{\displaystyle b\neq 1}" loading="lazy">. This means that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\neq 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./32a55c3a4b8eca743088b70c7c4d63c773f43c2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.258ex; height:2.676ex;" alt="{\displaystyle b\neq 1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a^{n}}{b^{n}}}}">

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6fafb316a65779077635b65898ad30f07f45c363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.284ex; height:5.343ex;" alt="{\displaystyle {\frac {a^{n}}{b^{n}}}}" loading="lazy"> is not in simplest form. Thus b should equal 1.

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{n}=1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./93f80fba3de21e4d0bae42b0e9bbc88f3abbdcdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.642ex; height:2.343ex;" alt="{\displaystyle 1^{n}=1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n}{1}}=n}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4b9c183ef2f1ffc5f7f09b3ec7c3d7ade7af9cb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.724ex; height:4.676ex;" alt="{\displaystyle {\frac {n}{1}}=n}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a^{n}}{b^{n}}}=a^{n}}">

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./05b2df4ed68facd9cc6e41861103dfdcf5dff2df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.831ex; height:5.343ex;" alt="{\displaystyle {\frac {a^{n}}{b^{n}}}=a^{n}}" loading="lazy">.

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

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./407e45335787b25bf78954908ec685cfb6b23622.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.876ex; height:2.343ex;" alt="{\displaystyle x=a^{n}}" loading="lazy"> and thus, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}=a}">

<mroot>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./39c0048b93aee48d4f00d14b120a98c1fbbcc67d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.594ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{x}}=a}" loading="lazy">. This implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}}">

<mroot>

</mrow>

</mroot>

</mrow>

</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"> is an integer. Since x is not a perfect nth power, this is impossible. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}}">

<mroot>

</mrow>

</mroot>

</mrow>

</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"> is irrational.

<ul><li><a href="Geometric_mean" title="Geometric mean">Geometric mean</a></li>

<li><a href="Twelfth_root_of_two" title="Twelfth root of two">Twelfth root of two</a></li></ul>

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

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<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.merriam-webster.com/dictionary/radication">"Definition of RADICATION"</a>. www.merriam-webster.com.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFMiller" class="citation web cs1">Miller, Jeff. <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/s.html">"Earliest Known Uses of Some of the Words of Mathematics"</a>. Mathematics Pages. Retrieved 2008-11-30.</cite>

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFHardy1921" class="citation book cs1"><a href="G._H._Hardy" title="G. H. Hardy">Hardy, G. H.</a> (1921). <a rel="nofollow" class="external text" href="https://archive.org/details/coursepuremath00hardrich/page/n36/mode/2up">A Course of Pure Mathematics</a> (3rd ed.). Cambridge. §1.13 "Quadratic Surds" – §1.14, pp. 19–23.</cite>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFMcKeague2011" class="citation book cs1">McKeague, Charles P. (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=etTbP0rItQ4C&q=editions:q0hGn6PkOxsC">Elementary algebra</a>. Cengage Learning. p. 470. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8400-6421-9</bdi>.</cite>

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFCavinessFateman" class="citation conference cs1">Caviness, B. F.; Fateman, R. J. <a rel="nofollow" class="external text" href="http://www.eecs.berkeley.edu/~fateman/papers/radcan.pdf">"Simplification of Radical Expressions"</a> (PDF). Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation. p. 329.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFRichard1985" class="citation journal cs1">Richard, Zippel (1985). "Simplification of Expressions Involving Radicals". Journal of Symbolic Computation. 1 (189–210): 189–210. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0747-7171%2885%2980014-6">10.1016/S0747-7171(85)80014-6</a>.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFWantzel1837" class="citation cs1">Wantzel, <a href="Monsieur" title="Monsieur">M.</a> L. (1837). <a rel="nofollow" class="external text" href="http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16381&Deb=374&Fin=380&E=PDF">"Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas"</a>. Journal de Mathématiques Pures et Appliquées. 1 (2): 366–372.</cite>

</li>

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<ul><li><a href="Primitive_recursive_function#Predecessor" title="Primitive recursive function">Predecessor (0)</a></li>

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<ul><li><a href="Primitive_recursive_function#Predecessor" title="Primitive recursive function">Predecessor (0)</a></li>

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<li><a href="Division_(mathematics)" title="Division (mathematics)">Division (2)</a></li>

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</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>

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