Exponent Combination Laws

Theorem
Let $a, b \in \R_+$ be positive real numbers.

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

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

Then:
 * $a^{x + y} = a^x a^y$


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


 * $a^{-x} = \dfrac 1 {a^x}$


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


 * $\dfrac{a^x}{a^y} = a^{x-y}$


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


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

Proof
We use the definition $a^x = \exp \left({x \ln a}\right)$ throughout.


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


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


 * $a^{-x} = \dfrac 1 {a^x}$:


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


 * $\dfrac{a^x}{a^y} = a^{x-y}$:


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


 * $\left(\dfrac{a}{b}\right) ^{-x} = \left(\dfrac{b}{a}\right) ^x$.