Sum of Indices of Real Number/Rational Numbers

Theorem

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

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

Then:

$r^{x + y} = r^x \times r^y$

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

Then:

$\ds r^\paren {x + y}$

$=$

$\ds r^\paren {\paren {p / q} + \paren {u / v} }$

$\ds $

$=$

$\ds r^\paren {\paren {p v + u q} / q v}$

$\ds $

$=$

$\ds \paren {r^\paren {1 / q v} }^\paren {p v + u q}$

Definition of Rational Power

$\ds $

$=$

$\ds \paren {r^\paren {1 / q v} }^\paren {p v} \times \paren {r^\paren {1 / q v} }^\paren {u q}$

$\ds $

$=$

$\ds r^\paren {p v / q v} \times r^\paren {u q / q v}$

Definition of Rational Power

$\ds $

$=$

$\ds r^\paren {p / q} \times r^\paren {u / v}$

$\ds $

$=$

$\ds r^x \times r^y$

$\blacksquare$