# Definition:Root (Analysis)

## Definition

Let $x, y \in \R_{\ge 0}$ be positive real numbers.

Let $n \in \Z$ be an integer such that $n \ne 0$.

Then $y$ is the positive $n$th root of $x$ if and only if:

$y^n = x$

and we write:

$y = \sqrt[n] x$

Using the power notation, this can also be written:

$y = x^{1/n}$

When $n = 2$, we write $y = \sqrt x$ and call $y$ the positive square root of $x$.

When $n = 3$, we write $y = \sqrt [3] x$ and call $y$ the cube root of $x$.

Note the special case where $x = 0 = y$:

$0 = \sqrt [n] 0$

## Also known as

A root is also known as a radical.

The operation of extracting a root of a number is also seen in some sources referred to as evolution.

By a similar naming convention, raising a number to a power is seen the same sources referred to as involution.

## Also see

• Results about roots of numbers can be found here.