> Exponentiation to zero annoyed me as a kid

Your teacher didn't tell you (or told you, but you didn't recognize it as something valuable and forgot it) that exponentiation to zero is a new definition over exponentiation to positive integers. Exponentiation to positive integers is defined somehow, and that definition says nothing about exponentiation to zero. It is a new definition, not something that you deduce.

The same holds for 0^0 or 0/0 (with some amounts of confusion, lies, and hypocriticism).