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Video transcript
So let's think a little bit about powers of 0. So what do you think 0 to the first power is going to be? And I encourage you to pause this video. Well, let's just think about it. One definition of exponentiation is that you start with a 1, and then, you multiply this number times a 1 one time. So this is literally going to be 1 times-- let me do it in the right color-- it's 1 times 0. You're multiplying the 1 by 0 one time. 1 times 0, well, that's just going to be equal to 0. Now, what do you think 0 squared or 0 to the second power is going to be equal to? Well, once again, one way of thinking about this is that you start with a 1, and we're going to multiply it by 0 two times. So times 0 times 0. Well, what's that going to be? Well, you multiply anything times 0, once again, you are going to get 0. And I think you see a pattern here. If I take 0 to any non-zero number-- so to the power of any non-zero, so this is some non-zero number, then this is going to be equal to 0. Now, this raises a very interesting question. What happens at 0 to the 0-th power? So here, 0 to the millionth power is going to be 0. 0 to the trillionth power is going to be 0. Even negative or fractional exponents, which we haven't talked about yet, as long as they're non-zero, this is just going to be equal to 0, kind of makes sense. But now, let's think about what 0 to the 0-th power is, because this is actually a fairly deep question. And I'll give you a hint. Well, actually, why don't you pause the video and think a little bit about what 0 to the 0-th power should be. Well, there's two trains of thought here. You could say, look, 0 to some non-zero number is 0. So why don't we just extend this to all numbers and say 0 to any number should be 0. And so maybe you should say that 0 to the 0-th power is 0. But then, there was another train of logic that we've already learned, that any non-zero number, if you take any non-zero number, and you raise it to the 0-th power. We've already established that you start with a 1, and you multiply it times that non-zero number 0 times. So this is always going to be equal to 1 for non-zero numbers. So maybe say, hey, maybe we should extend this to all numbers, including 0. So maybe 0 to the 0-th power should be 1. So we could make the argument that 0 to the 0-th power should be equal to 1. So you see a conundrum here, and there's actually really good cases, and you can get actually fairly sophisticated with your mathematics. And there's really good cases for both of these, for 0 to 0-th being 0, and 0 to the 0-th power being 1. And so when mathematicians get into this situation, where they say, well, there's good cases for either. There's not a completely natural one. Either of these definitions would lead to difficulties in mathematics. And so what mathematicians have decided to do is, for the most part-- and you'll find people who will dispute this; people will say, no, I like one more than the other-- but for the most part, this is left undefined. 0 to the 0-th is not defined by at least just kind of more conventional mathematics. In some use cases, it might be defined to be one of these two things. So 0 to any non-zero number, you're going to get 0. Any non-zero number to the 0-th power, you're going to get 1. But 0 to the 0, that's a little bit of a question mark.