Definition:Root (Analysis)

Definition
Let $x \in \R$ be a real number such that $x > 0$.

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

Then there always exists a unique $y \in \R: y > 0$ such that $y^m = x$.

This $y$ is called the $m$th root of $x$, and is denoted $y = \sqrt [m] x$.

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

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

The $m$th root of $x$ can also be written, using the power notation, as $x^{1/m}$.

Note the special case where $x = 0$: $\sqrt [m] 0 = 0$.

Also see

 * Power.