\begin{definition}[Definition:Power (Algebra)/Rational Number]
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:
:$x^r = x^{p/q} = \paren {\sqrt [q] x}^p = \sqrt [q] {\paren {x^p} }$.
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.
\end{definition}