nth root

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an nth root of a <a href="/facts/Number/auQSAE2m">number</a> x is a number r which, when <a href="/facts/Exponentiation/pac6QY2g">raised to the power</a> of n, yields x: 
 
 
 
 
 r
 
 n
 
 
 =
 
 
 
 
 r
 ×
 r
 ×
 ⋯
 ×
 r
 
 ⏟
 
 
 
 n
 
  factors
 
 
 
 =
 x
 .
 
 
 {\displaystyle r^{n}=\underbrace {r\times r\times \dotsb \times r} _{n{\text{ factors}}}=x.}

The <a href="/facts/Positive_integer/u65og8JE">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="/facts/Square_root/AfrzfBdQ">square root</a> and a root of degree 3, a <a href="/facts/Cube_root/c8klHLFb">cube root</a>. Roots of higher degree are referred by using <a href="/facts/Ordinal_numeral/Lg6cCHuN">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 
 
 
 
 
 
 x
 
 n
 
 
 
 
 
 {\displaystyle {\sqrt[{n}]{x}}}
 
 using the <a href="/facts/Radical_symbol/jfKTFjtG">radical symbol</a> 
 
 
 
 
 
 
 
 x
 
 
 
 
 
 
 {\displaystyle {\sqrt {\phantom {x}}}}
 
. The square root is usually written as ⁠
 
 
 
 
 
 x
 
 
 
 
 {\displaystyle {\sqrt {x}}}
 
⁠, with the degree omitted. Taking the nth root of a number, for fixed ⁠
 
 
 
 n
 
 
 {\displaystyle n}
 
⁠, is the <a href="/facts/Inverse_function/Tg5nnXlu">inverse</a> of raising a number to the nth power, and can be written as a <a href="/facts/Fraction_(mathematics)/ghnQE9B3">fractional</a> exponent:

 
 
 
 
 
 x
 
 n
 
 
 
 =
 
 x
 
 1
 
 /
 
 n
 
 
 .
 
 
 {\displaystyle {\sqrt[{n}]{x}}=x^{1/n}.}

For a positive real number x, 
 
 
 
 
 
 x
 
 
 
 
 {\displaystyle {\sqrt {x}}}
 
 denotes the positive square root of x and 
 
 
 
 
 
 x
 
 n
 
 
 
 
 
 {\displaystyle {\sqrt[{n}]{x}}}
 
 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="/facts/Imaginary_number/6Xh9W3hq">imaginary</a> square roots, ⁠
 
 
 
 +
 i
 
 
 x
 
 
 
 
 {\displaystyle +i{\sqrt {x}}}
 
⁠ and ⁠
 
 
 
 −
 i
 
 
 x
 
 
 
 
 {\displaystyle -i{\sqrt {x}}}
 
⁠, where i is the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a>.
In general, any non-zero <a href="/facts/Complex_number/2w7mqI7e">complex number</a> has n distinct complex-valued nth roots, equally distributed around a complex circle of constant <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a>. (The nth root of 0 is zero with <a href="/facts/Multiple_root/dndUyetA">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="/facts/Multivalued_function/WH5ua6BL">multivalued function</a>. By convention the <a href="/facts/Principal_value/bCHR0LmD">principal value</a> of this function, called the principal root and denoted ⁠
 
 
 
 
 
 x
 
 n
 
 
 
 
 
 {\displaystyle {\sqrt[{n}]{x}}}
 
⁠, 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="/facts/Imaginary_part/2w7mqI7e">imaginary part</a>. The principal root of a positive real number is thus also a positive real number. As a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a>, the principal root is <a href="/facts/Continuous_function/sKbl02pB">continuous</a> in the whole <a href="/facts/Complex_plane/vE0QrxJp">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 or a radical. 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="/facts/Transcendental_functions/yMkeq4Eo">transcendental functions</a> or <a href="/facts/Transcendental_numbers/IrB7JppM">transcendental numbers</a> it is called an <a href="/facts/Algebraic_expression/CK7D57Dm">algebraic expression</a>.

Roots are used for determining the <a href="/facts/Radius_of_convergence/4Un0ANfP">radius of convergence</a> of a <a href="/facts/Power_series/vYh6sTps">power series</a> with the <a href="/facts/Root_test/rUJKuqkl">root test</a>. The nth roots of 1 are called <a href="/facts/Roots_of_unity/EqH0ho1Y">roots of unity</a> and play a fundamental role in various areas of mathematics, such as <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, <a href="/facts/Theory_of_equations/aUwHT1SD">theory of equations</a>, and <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a>.

nth root open-in-new

nth root