What is 5 to the power of 0?

The zero power of any nonzero number has a defined value of 1.

For any n not equal to zero, n0 = 1

That this is correct can be demonstrated by the addition of powers, e.g. n2 times n3 = n (2+3)= n5, so for zero powers n2 times n0 = n(2+0) = n2, where n0 is then identity, or 1.