Product of Indices of Real Number/Integers

Theorem
Let $r \in \R_{> 0}$ be a positive real number. Let $n, m \in \Z$ be positive integers.

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

Then:


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

Proof
From Product of Indices of Real Number: Positive Integers, we have that:


 * $m \in \Z_{\ge 0}: \forall n \in \Z: \paren {r^n}^m = r^{n m}$

It remains to be shown that:


 * $\forall m \in \Z_{<0}: \forall n \in \Z: \paren {r^n}^m = r^{n m}$

As $m < 0$ we have that $m = -p$ for some $p \in \Z_{> 0}$.

Thus:

Hence the result, by replacing $-p$ with $m$.