Power Zero

Theorem: $\; a^0 = 1\;\;\forall\; a\neq 0$

Prerequisites:
Sum of the powers property (proof)

Proof:

$\qquad\quad \begin{align}a^0 &= a^{n-n} \qquad\quad\;\;\;\text{ where $n\in\mathbb{Z^+}$}\\ &= a^n.a^{-n}\qquad\quad\text{(by sum of the power property)}\\ &= \dfrac{a^n}{a^n} = 1\end{align}$

Hence the result


Recommended:
Sum of the powers
Product of the powers
Power of the product