Exponent Combination Laws/Power of Power/Proof 1

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Theorem

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


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

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


Then:

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


Proof

\(\ds a^{x y}\) \(=\) \(\ds \map \exp {x y \ln a}\) Definition of Power to Real Number
\(\ds \) \(=\) \(\ds \map \exp {y \, \map \ln {a^x} }\) Logarithms of Powers
\(\ds \) \(=\) \(\ds \paren {a^x}^y\) Definition of Power to Real Number

$\blacksquare$


Sources