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:

Product of Powers

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

Power of Product

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

Negative Power

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

Power of Power

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

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

Then:

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

Quotient of Powers

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

Power of Quotient

$\paren {\dfrac a b}^x = \dfrac {a^x} {b^x}$

Negative Power of Quotient

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