# Exponent Combination Laws/Quotient of Powers

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## Theorem

Let $a \in \R_{>0}$ be a positive real number.

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

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

Then:

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

## Proof

 $\ds \frac {a^x} {a^y}$ $=$ $\ds a^x \paren {\frac 1 {a^y} }$ $\ds$ $=$ $\ds \paren {a^x} \paren {a^{-y} }$ Exponent Combination Laws: Negative Power $\ds$ $=$ $\ds a^{x - y}$ Product of Powers

$\blacksquare$