Why does 5 to no power equal 1?

Answer 1

It doesn'tFive to "no power" is, by convention, the same as five raised to the power of one (51), which equals five.

Five to the power of zero (50), on the other hand, is, by convention, equal to one, because any number raised to the power of zero is one. Don't try too hard to conceptualize a number raised to the power of zero. It evaluates to one merely because that is the convention.

I'm sure that's what you meant when you said "no power."

Just a convention?I would disagree with the above answer and say that it's more than a convention. Some of the rules for exponentiation may seem strange at first (eg 5-1, 51/2). But if you change the rules, then axioms such as xa+b = xa * xb no longer hold. That means that if you ever do a proof involving these axioms, you get bogged down in a case analysis, i.e. different proofs for different values of x. So, while you are of course welcome to invent your own functions, they won't be as useful as the versions that are already defined.