Exponent Combination Laws/Power of Power

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Theorem

Let $a \in \R_+$ 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:

$\left({a^x}\right)^y = a^{xy}$

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

$\blacksquare$

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