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 $x$ :
 * $y^n = x$

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 see

 * Existence and Uniqueness of Positive Root of Positive Real Number, which proves the existence and uniqueness of the positive $n$th root


 * Definition:Power (Algebra)


 * Definition:Root of Unity