# Exponent Combination Laws/Negative Power

## Theorem

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

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

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

Then:

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

## Proof

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle a^{-x}$$ $$=$$ $$\displaystyle$$ $$\displaystyle \exp \left({-x \ln a}\right)$$ $$\displaystyle$$ $$\displaystyle$$ Definition of Power to Real Number $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \left({\exp \left({x \ln a}\right)}\right)^{-1}$$ $$\displaystyle$$ $$\displaystyle$$ Exponent of Product $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \frac 1 {\exp \left({x \ln a}\right)}$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \frac 1 {a^x}$$ $$\displaystyle$$ $$\displaystyle$$ Definition of Power to Real Number

$\blacksquare$