Exponent Combination Laws/Power of Power/Proof 1
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Theorem
Let $a \in \R_{>0}$ be a (strictly) positive real number.
Let $x, y \in \R$ be real numbers.
Let $a^x$ be defined as $a$ to the power of $x$.
Then:
- $\paren {a^x}^y = a^{x y}$
Proof
\(\ds a^{x y}\) | \(=\) | \(\ds \map \exp {x y \ln a}\) | Definition of Power to Real Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \exp {y \, \map \ln {a^x} }\) | Logarithms of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a^x}^y\) | Definition of Power to Real Number |
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 14.7 \ (1) \ \text{(iv)}$