Nth root

In mathematics, an nth root of a number x, where n is usually assumed to be a positive integer, is a number r which, when raised to the power n yields x

where n is the degree of the root. 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.

For example:

A real number or complex number has n complex roots of degree n. While the roots of 0 are not distinct (all equaling 0), the n nth roots of any other real or complex number are all distinct. If n is even and x is real and positive, one of its nth roots is positive, one is negative, and the rest are either non-existent (in the case when n = 2) or complex but not real; if n is even and x is real and negative, none of the nth roots is real. If n is odd and x is real, one nth root is real and has the same sign as x , while the other roots are not real. Finally, if x is not real, then none of its nth roots is real.

Roots are usually written using the radical symbol or radix ${\displaystyle {\sqrt {\,\,}}}$ or ${\displaystyle \surd {}}$, with ${\displaystyle {\sqrt {x}}\!\,}$ or ${\displaystyle \surd x}$ denoting the square root, ${\displaystyle {\sqrt[{3}]{x}}\!\,}$ denoting the cube root, ${\displaystyle {\sqrt[{4}]{x}}}$ denoting the fourth root, and so on. In the expression ${\displaystyle {\sqrt[{n}]{x}}}$, n is called the index, ${\displaystyle {\sqrt {\,\,}}}$ is the radical sign or radix, and x is called the radicand. Since the radical symbol denotes a function, when a number is presented under the radical symbol it must return only one result, so a non-negative real root, called the principal nth root, is preferred rather than others.

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