Exponent Combination Laws

Theorem

Let $a, b \in \R_{>0}$ be strictly positive real numbers.

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

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

Then:

Sum of Powers

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

Power of Product

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

Negative Power

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

Power of Power

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

Difference of Powers

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

Power of Quotient

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

Negative Power of Quotient

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

Also see

• Powers of Group Elements