introducing 0th power

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 $n$ . Using the fact that the integer 1 is a multiplicative identity, ($a\cdot 1=a$ for any $a$), we can write

$$a^n\cdot 1=a^n=a^{n+0}=a^n\cdot a^0,$$

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.