Definition:Root of Number

From ProofWiki
Jump to navigation Jump to search


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 $x = 0 = y$:

$0 = \sqrt [n] 0$


Let $\sqrt [n] x$ denote the $n$th root of $x$.

The number $n$ is known as the index of the root.

If $n$ is not specified, that is $\sqrt x$ is presented, this means the square root.

Extraction of Root

The process of evaluating roots of a given real number is referred to as extraction.

Also known as

A root of a number is also known as a radical.

Also see

  • Results about roots of numbers can be found here.