exponentiation - 0's Exponents are impossible?

I've had something that's been bugging me, and I tried research and asked my math teacher. None had sufficient answers. The concept of $0$ is that when $0$ goes to any exponent except for $0$, it becomes $0$. For example,

$0^3 = 0$, but $0^0 =$ undefined

However, the proof that $0^0$ is undefined is shown thus: $0^x$ ... (divided by) = $0^{(x-x)} = 0^0$ = undefined $0^x$

You can apply this to any exponent though, such as: $0^6$ ... = $0^6 = 0$ and $0^3 = 0$, so this expression is equal to $0/0$, which should be $0^3$ undefined, right?

Am I doing something wrong here? Please help! Gil

Answer

One should not say "equals undefined"; one should say "is undefined". The is the "is" of predication, not the "is" of equality.

$0^0$ is indeterminate in the sense that if $f(x)$ and $g(x)$ both approach $0$ as $x\to a$ then $f(x)^{g(x)}$ could approach any positive number or $0$ or $\infty$ depending on which functions $f$ and $g$ are.

But in some contexts it is important that $0^0$ be taken to be $1$. One of those contexts is in the identity $$e^z = \sum_{n=0}^\infty \frac{z^n}{n!}.$$ This fails to hold when $z=0$ unless $0^0=1$, since the first term of the series is then $\dfrac{0^0}{0!}$.

In combinatorial enumeration it is also important in some contexts that $0^0$ is $1$, for the same reason $2^0$ is $1$: if you multiply by something $0$ times, it's the same as multiplying by $1$. That is also one way of looking at the reason why $0!$ is $1$.