Question

Why can't you have zero to the power of zero?

Answer 1

This is a really good question. In general, and in most situations, mathematicians define `0^0 = 1`.

But that is the short answer. This question has been debated since the time of Euler (i.e. hundreds of years.)

We know that any nonzero number raised to the `0` power equals `1`
`n^0 = 1`

And that zero raised to a nonzero number equals `0`
`0^n = 0`

Sometime `0^0` is defined as indeterminate, that is in some cases it seems to be equal to `1` and others `0.`

Two source I used are:
http://mathforum.org/dr.math/faq/faq.0.to.0.power.html

https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-negative-numbers-multiply-and-divide/cc-7th-exponents-negative-base/v/powers-of-zero

Answer 2

Well, you kind of could have `0^0`. In general, mathematicians leave `0^0` undefined. There are 3 considerations that might lead someone to set a definition for `0^0`.
The problem (if it is a problem) is that they don't agree on what the definition should be.

Consideration 1:
For any number `p` other than `0`, we have `p^0=1`.

This is actually a definition of what the zero exponent mean. It's a definition chosen for good reasons. (And it doesn't "break" arithmetic.)

Here's one of the good reasons: defining `p^0` to be `1` lets us keep (and extend) the rules for working with exponents,
For example, `(5^7)/(5^3)=5^4` This works by cancellation and also by the rule `(p^n)/(p^m)=p^(n-m)` for `n>m`.
So what about `(5^8)/(5^8)`?
Cancellation (reducing the fraction) gives us `1`. We get to keep our "subtract the exponents" rule if we define `5^0` to be `1`.
So, maybe we should use the same rule to define `0^0`.
But . . .

Consideration 2
For any positive exponent, `p`, we have `0^p=0`. (This is not a definition, but a fact we can prove.)
So if it's true for positive exponants, maybe we should extend it to the `0` exponent and define `0^0=0`.

Consideration 3
We have looked at the expressions: `x^0` and `0^x`.
Now look at the expression `x^x`. Here's the graph of `y=x^x`:

graph{y=x^x [-1.307, 3.018, -0.06, 2.103]}

One of the things you may notice about this, is that when `x` is very close to `0` (but still positive), `x^x` is very close to `1`.

In some fields in mathematics, this is good reason to define `0^0` to be `1`.

Final notes
Definition is important and powerful, but cannot be used carelessly. I mentioned "breaking arithmetic". Any attempt to define division so that division by `0` is allowed will break some important part of arithmetic. Any attempt.

Last note: the definitions of `x^(-n)=1/(x^n)` and `x^(1/n)= root(n)x` are also motivated in part, by a desire to keep our familiar rules for working with exponents.