Exponent Combination Laws/Positive Integers

Theorem

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

Sum of Indices

Let $n, m \in \Z_{\ge 0}$ be positive integers.

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

Then:

$r^{n + m} = r^n \times r^m$

Power of Power

Let $n, m \in \Z_{\ge 0}$ be positive integers.

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

Then:

$\paren {r^n}^m = r^{n m}$