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

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

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.