Current time:0:00Total duration:4:15
0 energy points
Any non-zero number to the zero power equals one. Zero to any positive exponent equals zero. So, what happens when you have zero to the zero power? Created by Sal Khan.
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.