# Exponent Combination Laws/Power of Product

## Theorem

Let $a, b \in \R_{\ge 0}$ be positive real numbers.

Let $x \in \R$ be a real number.

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

Then:

$\paren {a b}^x = a^x b^x$

## Proof

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

$\blacksquare$