# Product of Indices of Real Number/Rational Numbers

## Theorem

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

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

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

Then:

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

## Proof

Let $x = \dfrac p q, y = \dfrac u v$.

Consider $\paren {\paren {r^x}^y}^{q v}$.

Then:

 $\ds \paren {\paren {r^x}^y}^{q v}$ $=$ $\ds \paren {\paren {r^\paren {p / q} }^\paren {u / v} }^{q v}$ $\ds$ $=$ $\ds \paren {\paren {\paren {r^\paren {p / q} }^\paren {u / v} }^v}^q$ Product of Indices of Real Number: Integers $\ds$ $=$ $\ds \paren {\paren {r^\paren {p / q} }^u}^q$ Definition of Rational Power $\ds$ $=$ $\ds \paren {r^\paren {p / q} }^{q u}$ Product of Indices of Real Number: Integers $\ds$ $=$ $\ds \paren {\paren {r^\paren {p / q} }^q}^u$ Product of Indices of Real Number: Integers $\ds$ $=$ $\ds \paren {r^p}^u$ Definition of Rational Power $\ds$ $=$ $\ds r^{p u}$ Product of Indices of Real Number: Integers $\ds \leadsto \ \$ $\ds \paren {\paren {r^\paren {p / q} }^\paren {u / v} }$ $=$ $\ds r^{\paren {p u} / \paren {q v} }$ taking $q v$th root of both sides $\ds \leadsto \ \$ $\ds \paren {r^x}^y$ $=$ $\ds r^{x y}$

$\blacksquare$