Exponent Combination Laws/Quotient of Powers

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

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Laws of Exponents: $7.2$
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): exponent (index)
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): exponent (index)
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 13$: Laws of Exponents: $13.2.$
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): index (indices) (ii)
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