Let $a$ be a number not equal to zero. Then for all $n\in\mathbb{N}$, we have that $a^{n}$ is the product of $a$ with itself $n$ times. Using the fact that the integer 1 is a multiplicative identity, ($a\cdot 1=a$ for any $a$), we can write

where we have used the properties of exponents under multiplication. Now, after canceling a factor of $a^{n}$ from both sides of the above equation, we derive that $a^{0}=1$ for any non-zero number.