Zeroth Power of Real Number equals One
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Theorem
Let $a \in \R_{>0}$ be a (strictly) positive real number.
Let $a^x$ be defined as $a$ to the power of $x$.
Then:
- $a^0 = 1$
Proof
\(\ds a^0\) | \(=\) | \(\ds \map \exp {0 \ln a}\) | Definition of Power to Real Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \exp 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | Exponential of Zero |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Laws of Exponents: $7.4$