Product of Indices of Real Number/Integers

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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:

\(\displaystyle \paren {r^n}^{- p}\) \(=\) \(\displaystyle \dfrac 1 {\paren {r^n}^p}\) Real Number to Negative Power: Integer
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {r^{n p} }\) from $(1)$
\(\displaystyle \) \(=\) \(\displaystyle r^{-n p}\) Real Number to Negative Power: Integer

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

$\blacksquare$


Sources