A powered by B equals B powered by A. A does not equal B. What's A and what's B?

Step-by-step explanation:

I know that there is a zero and a 1 involved in this somewhere. The only answer I can get which is a bit of a cheat, is

1^0! = 0!^1

These are not exactly the same thing. The trouble is that they are equal. There is a difference between "not the same thing" and "equal."

If I think of something else, I'll put it in a comment.