Product of Indices of Real Number/Rational Numbers

Theorem
Let $r \in \R_{> 0}$ be a (strictly) positive real number. Let $x, y \in \Q$ be rational numbers.

Let $r^x$ be defined as $r$ to the power of $x$.

Then:


 * $\paren {r^x}^y = r^{x y}$

Proof
Let $x = \dfrac p q, y = \dfrac u v$.

Consider $\paren {\paren {r^x}^y}^{q v}$.

Then: