Zero Exponent Rule

Here's one simple explanation of why the zeroth power of any non-zero value equals $\hspace{0.2em} 1 \hspace{0.2em}$.

The quotient rule for exponents tell us —

Now, let's see what happens if we replace $\hspace{0.2em} 0 \hspace{0.2em}$ with $\hspace{0.2em} m - m \hspace{0.2em}$ in $\hspace{0.2em} a^0 \hspace{0.2em}$.

See why $\hspace{0.2em} a^m = 1 \hspace{0.2em}$? Now, of course, $\hspace{0.2em} a \hspace{0.2em}$ can't be $\hspace{0.2em} 0 \hspace{0.2em}$ because in the second step, we are dividing by $\hspace{0.2em} a^m \hspace{0.2em}$. And we can't divide by $\hspace{0.2em} 0 \hspace{0.2em}$.

Let me explain briefly what makes the case of $\hspace{0.2em} 0^0 \hspace{0.2em}$so special.

We have two different exponent rules that give two different values for $\hspace{0.2em} 0^0$.

One, $\hspace{0.2em} 0 \hspace{0.2em}$ raised to any non-zero exponent is $\hspace{0.2em} 0 \hspace{0.2em}$.

If we extend this rule to $\hspace{0.2em} 0^0$, it's value would be $\hspace{0.2em} 0 \hspace{0.2em}$.

Two, the zero exponent rule. Any non-zero base raised to the zeroth power is $\hspace{0.2em} 0 \hspace{0.2em}$.

Now, if we extend this rule to $\hspace{0.2em} 0^0$, it's value would be $\hspace{0.2em} 1 \hspace{0.2em}$.

So you see, which of the two values do we go with — $\hspace{0.2em} 0 \hspace{0.2em}$ or $\hspace{0.2em} 1 \hspace{0.2em}$? Now, most people side with $\hspace{0.2em} 1 \hspace{0.2em}$ because the second rule is more consequential. And, hence, more important to remain consistent with.