Definition:Power (Algebra)/Rational Number

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

Let $m \in \Z$ be an integer.

Let $y = \sqrt [m] x$ be the $m$th root of $x$.

Then we can write $y = x^{1/m}$ which means the same thing as $y = \sqrt [m] x$.

Thus we can define the power to a positive rational number:

Let $r = \dfrac p q \in \Q$ be a positive rational number where $p \in \Z_{\ge 0}, q \in \Z_{> 0}$.

Then $x^r$ is defined as:


 * $x^r = x^{p/q} = \left({\sqrt [q] x}\right)^p = \sqrt [q] {\left({x^p}\right)}$.

When $r = \dfrac {-p} q \in \Q: r < 0$ we define:


 * $x^r = x^{-p/q} = \dfrac 1 {x^{p/q}}$ analogously for the negative integer definition.

Also see

 * Definition:Power of Zero for the definition of $x^r$ where $x = 0$.

Historical Note
The definition:
 * $x^r = x^{p/q} = \left({\sqrt [q] x}\right)^p = \sqrt [q] {\left({x^p}\right)}$

is due to circa 1360 C.E.