# Exponent Combination Laws/Product of Powers/Proof 1

## Theorem

Let $a \in \R_{> 0}$ be a positive real number.

Let $x, y \in \R$ be real numbers.

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

Then:

$a^x a^y = a^{x + y}$

## Proof

 $\ds a^{x + y}$ $=$ $\ds \map \exp {\paren {x + y} \ln a}$ Definition of Power to Real Number $\ds$ $=$ $\ds \map \exp {x \ln a + y \ln a}$ $\ds$ $=$ $\ds \map \exp {x \ln a} \, \map \exp {y \ln a}$ Exponential of Sum $\ds$ $=$ $\ds a^x a^y$ Definition of Power to Real Number

$\blacksquare$