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Root (ⁿ√x) ○◂｜Definition｜1st｜20260122103308-00-⌔

nth root - Wikipedia

n th root

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In mathematics, an nth root of a number x is the number r which, when multiplied by itself n times, yields x:

$r^n=\underset{n\text{ factors}}{\underbrace{r\times r\times \cdots \times r}}=x.$

The positive integer 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 square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an n th root is a root extraction.

The n th root of x is written as $\sqrt[n]{x}$ using the radical symbol x​. The square root is usually written as ⁠ $\sqrt{x}$ ⁠, with the degree omitted. Taking the n th root of a number, for fixed ⁠ $n$ ⁠, is the inverse of raising a number to the n th power,¹ and can be written as a fractional exponent:

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

For a positive real number x, $\sqrt{x}$ denotes the positive square root of x and $\sqrt[n]{x}$ denotes the positive real n th root. For example, 3 is a square root of 9, since 3${}^2$ = 9, and −3 is also a square root of 9, since (−3)${}^2$ = 9.² A negative real number − x has no real-valued square roots, but when x is treated as a complex number it has two imaginary square roots, ⁠ $+i\sqrt{x}$ ⁠ and ⁠ $-i\sqrt{x}$ ⁠, where i is the imaginary unit.

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

An unresolved root, especially one using the radical symbol, is sometimes referred to as a surd³ or a radical.⁴ 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.

Printed 2026-06-28.

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Footnotes

1. “Lesson Explainer: nth Roots: Integers”. Retrieved 22 July 2023. ↩

2. Zuckerman, Martin M. (1986). Intermediate Algebra: A Straightforward Approach. Bloomsbury Publishing PLC. pp. 249–251. ISBN 978-1-4616-3783-7. ↩

3. Bansal, R. K. (2006). New Approach to CBSE Mathematics IX. Laxmi Publications. p. 25. ISBN 978-81-318-0013-3. ↩

4. Silver, Howard A. (1986). Algebra and trigonometry. Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 978-0-13-021270-2. ↩

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