Exponent Combination Laws/Positive Integers

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Theorem

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


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:

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