What is (0/0)^0?

A professional mathematician would answer, "That's just silly. It's called an indeterminate form."

Let's try anyway... (0/0)^0 = 0^0 / 0^0 ... So what is 0^0? Indeterminate... I would say if it is a number, it would be 1, but it probably isn't a number.

1

The above answer is correct.

When you learned division in elementary school, it would have been nice if the teacher explained that division is simply not permitted by zero. It is not a permitted operation. Putting a zero in the numerator or as an exponent does not change that.

Now, a different question might be to ask whether y= (x/x)^x, as x gets closer and closer to zero, approaches any finite number; but the answer would be different for y = ((x^2)/(x))^x.

This is undefined. You can't divide by 0. Anything else raised to the 0 is 1. 0 divided by anything (except 0) is 0.