# Exponent Combination Laws/Power of Quotient

From ProofWiki

## Theorem

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

Let $x \in \R$ be a real number.

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

Then:

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

## Proof

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \left({\frac a b}\right)^x\) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \exp \left({x \ln \left({\frac a b}\right)}\right)\) | \(\displaystyle \) | \(\displaystyle \) | Definition of Power to Real Number | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \exp \left({x \ln a - x \ln b}\right)\) | \(\displaystyle \) | \(\displaystyle \) | Sum of Logarithms | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \frac{\exp \left({x \ln a}\right)}{\exp \left({x \ln b}\right)}\) | \(\displaystyle \) | \(\displaystyle \) | Exponent of Sum | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \frac{a^x}{b^x}\) | \(\displaystyle \) | \(\displaystyle \) | Definition of Power to Real Number |

$\blacksquare$