Zeroth Power of Real Number equals One

From ProofWiki
Jump to navigation Jump to search

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