Powers of zero | Exponents, radicals, and scientific notation | Pre-algebra | Khan Academy

In this video, we're going to talk about powers of zero. Just as a little bit of a reminder, let's start with a non-zero number just to remind ourselves what exponentiation is all about.

So, if I were to take 2 to the first power, one way to think about this is we always start with a one, and then we multiply this base that many times times that one. So here we're going to have one, two. So it's going to be one times 2, which is of course equal to 2.

If I were to say, what is 2 to the second power? Well, that's going to be equal to 1 times, and now I'm going to have two twos. So, times 2 times 2, which is equal to 4. You could keep going like that.

Now, the reason why I have this 1 here, and we've done this before, is to justify, and there's many other good reasons why 2 to the zero power should be equal to one. But you could see if we use the same exact idea here: you start with a one, and then you multiply it by two zero times. Well, that's just going to end up with a one.

So, so far I've told you this video is about powers of zero, but I've been doing powers of two. So let's focus on zero now. What do you think zero to the first power is going to be? Pause this video and try to figure that out.

Well, you do the exact same idea: you start with a one and then multiply it by zero one time. So, times zero, and this is going to be equal to zero. What do you think zero to the second power is going to be equal to? Pause this video and think about that.

Well, it's going to be 1 times 0 twice. So, times 0 times 0, and I think you see where this is going. This is also going to be equal to zero. What do you think zero to some arbitrary positive integer is going to be?

Well, it's going to be equal to 1 times 0 that positive integer number of times. So, once again, it's going to be equal to 0. In general, you can extend that 0 to any positive value exponent; it's going to give you zero. So, that's pretty straightforward.

But there is an interesting edge case here. What do you think zero to the zeroth power should be? Pause this video and think about that.

Well, this is actually contested; different people will tell you different things. If you use the intuition behind exponentiation that we've been using in this video, you would say, all right, I would start with a one and then multiply it by zero zero times. Or in other words, I just wouldn't multiply it by zero, in which case I'm just left with the one.

That means zero to the zero power should be equal to one. Other folks would say, hey, no, I'm with a zero, and that's the zeroth power; maybe it should be a zero. That's why a lot of folks leave it undefined. Most of the time, you're going to see zero to the zero power either being undefined or that it is equal to 1.