Exponent Combination Laws/Power of Product

Theorem

Let $a, b \in \R_{\ge 0}$ 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:

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

$\ds \paren {a b}^x$

$=$

$\ds \map \exp {x \map \ln {a b} }$

Definition of Power to Real Number

$\ds $

$=$

$\ds \map \exp {x \ln a + x \ln b}$

$\ds $

$=$

$\ds \map \exp {x \ln a} \map \exp {x \ln b}$

$\ds $

$=$

$\ds a^x b^x$

Definition of Power to Real Number

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Laws of Exponents: $7.6$
1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 14.7 \ (1) \ \text{(ii)}$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 13$: Laws of Exponents: $13.6.$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): index (indices) (iv)