What is zero to the zeroth power?
Everything to the zeroth power is 1.
Undefined.
Well, if you look here, it seems mathematicians can’t quite agree either. Some argue that it should be 1, and it seems to have become the consensus now.
I kind of like this line from the article:
In other words, when we want it to be 1, it’ll be 1, and the rest of the time it will be indeterminate. 
Going by the normal definition of x[sup]y[/sup], there’s no good reason for 0[sup]0[/sup] to be any particular value. But if 0[sup]0[/sup] [symbol]¹[/symbol] 1, then (x + 0)[sup]1[/sup] can’t be expanded by the binomial theorem without making the process much more complicated than it is. That’s a good enough reason for me to want to define 0[sup]0[/sup] = 1.
On t’other hand, very few people suffer by not being able to expand (x + 0)[sup]1[/sup] by the binomial theorem. :dubious:
I’d also say it is indeterminate.
As someone pointed out, any number to the zeroth power is 1.
So by that reasoning, 0[sup]0[/sup] = 1
However, when zero is raised to any power, the answer is zero. For example 0[sup]2[/sup] = 0[sup]3[/sup] = 0[sup]4[/sup] =0
So by that reasoning, 0[sup]0[/sup] = 0
Seems indeterminate to me.
What do you mean not bothered? I might not get over it for seconds.
Another SDMB thread on the same subject:
I like indeterminate, mainly because that’s what I’ve always been told it was.
In calculus terms, 0[sup]0[/sup] is an indeterminate form: knowing that lim f(x) = 0 and lim g(x) = 0 doesn’t tell you what lim f(x)[sup]g(x)[/sup] has to be.
Which, translated into non-calculus terms, means that if you take something that’s really really close to 0, and raise it to a power that’s really really close to zero, the result could be really close to 0, or to 1, or to something else, depending on what the something and the power are. They’re going to fight over whether it should be 0 or 1, and who wins the fight (or whether there’s a compromise) depends on which one of them is stronger and more insistent.
Put it this way:
Have you ever graphed y=x^x?
Between x>0 and x=1, it makes a very nice bowl shape, with its sides at y=1. Visually speaking, then, it makes good sense to say that 0^0=1.
But it seems to be discontinuous at x=0, based on a freeware graphing utility that I found.
Graph y = 0[sup]x[/sup].
More generally it just breaks the rules. 0[sup]0[/sup] is in the same class as 0/0 – it just doesn’t make sense. Mathematics has a sort of grammar just like any other language, and “0[sup]0[/sup]” breaks those rules.
The limit of x^y as (x,y) approaches the origin along any straight-line path[sup]*[/sup] (other than the line x=0) is 1. Which is a large part of the reason why it’s usually more convenient to define 0^0 = 1. But there are other paths on which you can approach the origin, and on some of those paths, the limit is 0 or something else, so one can’t rigorously say that the limit is 1 (or even that the limit exists at all). So the discontinuity is not removable.
- I think this may even apply to any polynomial path, but I’m not certain about that.