Product of Indices of Real Number

Theorem

Let $r \in \R_{> 0}$ be a (strictly) positive real number.

Positive Integers

Let $n, m \in \Z_{\ge 0}$ be positive integers.

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

Then:

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

Integers

Let $n, m \in \Z$ be positive integers.

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

Then:

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

Rational Numbers

Let $x, y \in \Q$ be rational numbers.

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

Then:

$\paren {r^x}^y = r^{x y}$