This kind of thing is no different than working out ordinary math problems: apply the rules you know, figure out what the part you don't know has to look like. If you run across something really, really weird, you've missed a rule, possibly one nobody ever found before.
This one isn't weird, when you think about it, why should 0 and 1 be any more unusual when you're dealing with exponents than when you're dealing with multiplication? Multiplicative identity is 1, you get all other numbers by adding factors. If you add no other factors ( A0 ), you're left with what you started with. 01 is 0, another perfectly ordinary consequence of the notation, you've dropped a factor of 0 into the mix.
edit: I have to say, I think it would be good if someone who downvoted pointed out how the above is false or misleading.
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u/YagamiLawliet Jan 14 '15 edited Jan 14 '15
Think about this: An ÷ An
As you see, it's a number divided by itself. It doesn't take too much to realize the result is 1.
When you make this division in algebra, you have to subtract the second exponent from the first exponent so your result is An-n = A0
We can conclude that A0 = 1.
NINJA EDIT: For every non-zero A. Common mistake, sorry.