nth Root

nth Root

The nth root of a number x, where n is a positive integer, is a number r whose nth power is x:

Every positive real number x has a single positive nth root, which is written . For n equal to 2 this is called the square root and the n is omitted. The nth root can also be represented using exponentiation as x1/n.

For even values of n, positive numbers also have a negative nth root, while negative numbers do not have a real nth root. For odd values of n, every negative number x has a real negative nth root. For example, −2 has a real 5th root, but −2 does not have any real 6th roots.

Every non-zero number x, real or complex, has n different complex number nth roots including any positive or negative roots, see complex roots below. The nth root of 0 is 0.

For most numbers, an nth root is irrational. For example,

All nth roots of integers, or in fact of any algebraic number, are algebraic.

For the extension of powers and roots to indices that are not positive integers, see exponentiation.

The character codes for the radical symbols are