Editing Nth root (section)

{{short description|Arithmetic operation, inverse of nth power}}
{{about|nth-roots of real and complex numbers|other uses|Root (disambiguation)#Mathematics}}
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{{More citations needed|date=October 2022}}
In [[mathematics]], an '''{{mvar|n}}th root''' of a [[number]] {{mvar|x}} is a number {{mvar|r}} which, when [[exponentiation|raised to the power]] of {{mvar|n}}, yields&nbsp;{{mvar|x}}: <math display="block">r^n = \underbrace{r \times r \times \dotsb \times r}_{n\text{ factors}} = x.</math>

The [[positive integer]] {{mvar|n}} is called the ''index'' or ''degree'', and the number {{mvar|x}} of which the root is taken is the ''radicand.'' A root of degree 2 is called a ''[[square root]]'' and a root of degree 3, a ''[[cube root]]''. Roots of higher degree are referred by using [[ordinal numeral|ordinal numbers]], as in ''fourth root'', ''twentieth root'', etc.  The computation of an {{mvar|n}}th root is a '''root extraction'''.

For example, {{math|3}} is a square root of {{math|9}}, since {{math|1=3{{sup|2}} = 9}}, and {{math|−3}} is also a square root of {{math|9}}, since {{math|1=(−3){{sup|2}} = 9}}.

The {{mvar|n}}th root of {{mvar|x}} is written as <math>\sqrt[n]{x}</math> using the [[radical symbol]] <math>\sqrt{\phantom x}</math>. The square root is usually written as {{tmath|\sqrt x}}, with the degree omitted. Taking the {{mvar|n}}th root of a number, for fixed {{tmath|n}}, is the [[inverse function#Squaring and square root functions|inverse]] of raising a number to the {{mvar|n}}th power,<ref>{{cite web |url=https://www.nagwa.com/en/explainers/985195836913 |access-date=22 July 2023 |title=Lesson Explainer: nth Roots: Integers}}</ref> and can be written as a [[Fraction (mathematics)|fractional]] exponent:

<math display="block">\sqrt[n]{x} = x^{1/n}.</math>

For a positive real number {{mvar|x}}, <math>\sqrt{x}</math> denotes the positive square root of {{mvar|x}} and <math>\sqrt[n]{x}</math> denotes the positive real {{mvar|n}}th root. A negative real number {{math|−''x''}} has no real-valued square roots, but when {{mvar|x}} is treated as a complex number it has two [[imaginary number|imaginary]] square roots, {{tmath|+i\sqrt x }} and {{tmath|-i\sqrt x }}, where {{mvar|i}} is the [[imaginary unit]].

In general, any non-zero [[complex number]] has {{mvar|n}} distinct complex-valued {{mvar|n}}th roots, equally distributed around a complex circle of constant [[Absolute value#Complex numbers|absolute value]]. (The {{mvar|n}}th root of {{math|0}} is zero with [[multiple root|multiplicity]] {{mvar|n}}, and this circle degenerates to a point.) Extracting the {{mvar|n}}th roots of a complex number {{mvar|x}} can thus be taken to be a [[multivalued function]]. By convention the [[principal value]] of this function, called the '''principal root''' and denoted {{tmath|\sqrt[n]{x} }}, is taken to be the {{mvar|n}}th root with the greatest real part and in the special case when {{mvar|x}} is a negative real number, the one with a positive [[imaginary part]]. The principal root of a positive real number is thus also a positive real number. As a [[function (mathematics)|function]], the principal root is [[continuous function|continuous]] in the whole [[complex plane]], except along the negative real axis.

An unresolved root, especially one using the radical symbol, is sometimes referred to as a '''surd'''<ref>{{cite book |title=New Approach to CBSE Mathematics IX |first=R.K. |last=Bansal |page=25 |year=2006 |isbn=978-81-318-0013-3 |publisher=Laxmi Publications |url=https://books.google.com/books?id=1C4iQNUWLBwC&pg=PA25}}</ref> or a '''radical'''.<ref name=silver>{{cite book|last=Silver|first=Howard A.|title=Algebra and trigonometry|year=1986|publisher=Prentice-Hall|location=Englewood Cliffs, New Jersey|isbn=978-0-13-021270-2|url-access=registration|url=https://archive.org/details/algebratrigonome00silv}}</ref> 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 [[transcendental functions]] or [[transcendental numbers]] it is called an ''[[algebraic expression]]''.

{{Arithmetic operations}}

Roots are used for determining the [[radius of convergence]] of a [[power series]] with the [[root test]]. The {{mvar|n}}th roots of 1 are called [[roots of unity]] and play a fundamental role in various areas of mathematics, such as [[number theory]], [[theory of equations]], and [[Fourier transform]].