U+0607 Arabic-Indic Fourth Root

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In mathematics, an nth root of a number x is a number r (the root) which, when raised to the power of the positive integer n, yields x: $r^n=\underset{n\text{ factors}}{\underbrace{r\times r\times \cdots \times r}}=x.$

The 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 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 $\sqrt[n]{x}$ using the radical symbol or radix $\sqrt{x}$. The square root is usually written without the n as just ⁠$\sqrt{x}$⁠. Taking the nth root of a number is the inverse operation of exponentiation, 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 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 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 nth roots, equally distributed around a complex circle of constant absolute value. (The nth root of 0 is zero with multiplicity n, and this circle degenerates to a point.) Extracting the nth 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 nth 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.

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 transcendental functions or transcendental numbers it is called an algebraic expression.

Roots are used for determining the radius of convergence of a power series with the root test. The nth 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.