Real Number to Negative Power/Integer

Theorem
Let $r \in \R_{> 0}$ be a positive real number. Let $n \in \Z$ be an integer.

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

Then:


 * $r^{-n} = \dfrac 1 {r^n}$

Proof
Let $n \in \Z_{\ge 0}$.

Then from Real Number to Negative Power: Positive Integer:
 * $r^{-n} = \dfrac 1 {r^n}$

It remains to show that this holds when $n < 0$.

Let $n \in \Z_{<0}$.

Then $n = - m$ for some $m \in \Z_{> 0}$.

Thus: