Exponent Combination Laws/Quotient of Powers

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Theorem

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

Let $x, y \in \R$ be real numbers.

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


Then:

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


Proof

\(\ds \frac {a^x} {a^y}\) \(=\) \(\ds a^x \paren {\frac 1 {a^y} }\)
\(\ds \) \(=\) \(\ds \paren {a^x} \paren {a^{-y} }\) Exponent Combination Laws: Negative Power
\(\ds \) \(=\) \(\ds a^{x - y}\) Product of Powers

$\blacksquare$


Sources

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