nth Root

For a positive integer , an th root (or "th radical") of a quantity is a value such that . Thus taking an th root reverses the taking of an th power. Since different roots may exist, constructing a single-valued inverse function requires restricting the domain and choosing a branch. 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 nonzero complex number has exactly distinct th roots (Boyer 1968, p. 476). Since complex numbers have two square roots and three cube roots, care is needed in determining which root is under consideration. The principal root is obtained by using the complex argument in the range and taking the root whose complex argument is . When is odd and is a real number, applications may instead select the unique real root.

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

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

(1)

Writing

(2)

the roots are

(3)

where , 1, ..., . They have complex modulus

(4)

and complex arguments

(5)