Real Number to Negative Power/Integer

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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:

\(\ds r^{-m}\) \(=\) \(\ds \dfrac 1 {r^m}\) 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
\(\ds \) \(=\) \(\ds r^m\)
\(\ds \leadsto \ \ \) \(\ds \dfrac 1 {r^n}\) \(=\) \(\ds r^{-n}\)

$\blacksquare$