Exponent Combination Laws/Power of Product

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

$\blacksquare$