# 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

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