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:

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

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

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:

$\ds \paren {r^n}^{- p}$

$=$

$\ds \dfrac 1 {\paren {r^n}^p}$

$\ds $

$=$

$\ds \dfrac 1 {r^{n p} }$

from $(1)$

$\ds $

$=$

$\ds r^{-n p}$

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

$\blacksquare$