why does 0 to the power of 0 equal 1?
In: 97
What does it mean to take something to the power of zero?
Well, what does it mean to take something to a power *in general*?
a^n means multiply a by itself n times. So 2^3 is 2 x 2 x 2; 5^7 is 5 x 5 x 5 x 5 x 5 x 5 x 5.
What does a^0 mean, though? It means that you multiple something by itself…zero times? One way to interpret this is “What number, when multiplied by a once, yields a?”. The answer is 1. So mathematicians have decided on the rule that a^0 = 1. This is true regardless of what your a is. And this rule holds for 0 as well: 0^0 x 0 = 1 x 0 = 0, as expected.
This is the justification that lies behind the practice of 0^0 = 1. However, sometimes it’s also said that 0^0 is undefined; that is, mathematicians agree that 0^0 isn’t really a thing you can do.
The precise mathematical phrase 0^0 is not really applicable to real life; that means that the “true meaning” of the value is less important than whether the value is convenient (or easy or logical or whatever) to work with.
So, some mathematicians (mostly the ones that take weird things to weird powers a lot and want to be able to handle the edge cases with zero without having to write a special note at the bottom of their proofs) go with 1, some (mostly the ones who think it’s important to know precisely what statements can and cannot have meaning) go with undefined, because while 1 x 0 = 0, so does 2 x 0 and pi x 0 and -1,439,239.2384 x 0. And so it’s mostly a matter of usefulness and to some extent aesthetics.
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