nth Root

The th root (or "th radical") of a quantity is a value such that , and therefore is the inverse function to the taking of a power. The th root is denoted or, using power notation, . The special case of the square root () is denoted . The case is known as the cube root.

The quantities for which a general function equals 0 are also called roots, or sometimes zeros.

The quantities such that are called the th roots of unity.

Rolle proved that any complex number has exactly th roots (Boyer 1968, p. 476), though some are possibly degenerate. However, since complex numbers have two square roots and three cube roots, care is needed in determining which root is under consideration. For complex numbers , the root of interest (generally taken as the root having smallest positive complex argument) is known as the principal root. However, for real numbers, the root of interest is usually the root that is real (when it exists).

The principal th root of a complex number can be found in the Wolfram Language as z^(1/n) or equivalently Power[z, 1/n]. When only real roots are of interest, the command Surd[x, n] which returns the real-valued th root for real odd and the principal th root for nonnegative real and even can be used.

The th root of a complex number can be found analytically by solving the equation

(1)

Writing the th power of a complex number in terms of its norm and phase gives

			(2)
			(3)

so the roots have complex modulus

(4)

and complex argument

(5)