Exponent Combination Laws/Negative Power of Quotient

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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:

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

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \left({\frac a b}\right) ^{-x}$	$=$	$\displaystyle \left({\frac 1 {\left({\frac a b}\right)} }\right)^x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Exponent Combination Laws/Negative Power
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left({\frac b a}\right)^x$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\blacksquare$

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({\frac a b}\right) ^{-x}\)	\(=\)	\(\displaystyle \left({\frac 1 {\left({\frac a b}\right)} }\right)^x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Exponent Combination Laws/Negative Power
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({\frac b a}\right)^x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)