Exponent Combination Laws/Sum of Powers

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:

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle a^{x + y}$	$=$	$\displaystyle $	$\displaystyle \exp \left({\left({x + y}\right) \ln a}\right)$	$\displaystyle $	$\displaystyle $	Definition of Power to Real Number
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle $	$\displaystyle \exp \left({x \ln a + y \ln a}\right)$	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle $	$\displaystyle \exp \left({x \ln a}\right) \exp \left({y \ln a}\right)$	$\displaystyle $	$\displaystyle $	Exponent of Sum
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle $	$\displaystyle a^x a^y$	$\displaystyle $	$\displaystyle $	Definition of Power to Real Number

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle a^{x + y}\)	\(=\)	\(\displaystyle \)	\(\displaystyle \exp \left({\left({x + y}\right) \ln a}\right)\)	\(\displaystyle \)	\(\displaystyle \)	Definition of Power to Real Number
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle \exp \left({x \ln a + y \ln a}\right)\)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle \exp \left({x \ln a}\right) \exp \left({y \ln a}\right)\)	\(\displaystyle \)	\(\displaystyle \)	Exponent of Sum
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle a^x a^y\)	\(\displaystyle \)	\(\displaystyle \)	Definition of Power to Real Number