Exponent Combination Laws/Negative Power

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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          

$\blacksquare$


Sources