Exponent Combination Laws/Negative Power of Quotient

Theorem

Let $a, b \in \R_{>0}$ be (strictly) positive real numbers.

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

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

Then:

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

Proof

\(\ds \paren {\frac a b} ^{-x}\)

\(=\)

\(\ds \paren {\frac 1 {\paren {\frac a b} } }^x\)

Exponent Combination Laws: Negative Power

\(\ds \)

\(=\)

\(\ds \paren {\frac b a}^x\)

$\blacksquare$