Exponent Combination Laws/Power of Product

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Theorem

Let $a, b \in \R_+$ 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:

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


Proof

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

$\blacksquare$


Sources