# 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 $n \in \Z_{\ge 0}$.

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

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:

 $\ds r^{-m}$ $=$ $\ds \dfrac 1 {r^m}$ Real Number to Negative Power: Positive Integer $\ds \leadsto \ \$ $\ds \dfrac 1 {r^{-m} }$ $=$ $\ds \dfrac 1 {1 / r^m}$ taking reciprocal of both sides $\ds$ $=$ $\ds r^m$ $\ds \leadsto \ \$ $\ds \dfrac 1 {r^n}$ $=$ $\ds r^{-n}$

$\blacksquare$