Sum 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 $n$.

Then:


 * $r^{x + y} = r^x \times r^y$

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

Then: