Product of Indices of Real Number
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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:
- $\paren {r^x}^y = r^{x y}$