Exponent Combination Laws/Rational Numbers

Theorem

Let $r \in \R_{> 0}$ be a (strictly) positive real number.

Sum of Indices

Let $x, y \in \Q$ be rational numbers.

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

Then:

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

Power of Power

Let $x, y \in \Q$ be rational numbers.

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

Then:

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