Main content

## Exponent properties (integer exponents)

# Powers of zero

CCSS.Math:

## Video transcript

- [Instructor] In this video, we're going to talk about powers of zero. And just as a little bit of a reminder, let's start with a nonzero number, just to remind ourselves what
exponentiation is all about. So if I were to take
two 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 only gonna have one two, so it's gonna be one times two, which is, of course, equal to two. If I were to say what is
two to the second power? Well, that's going to
be equal to one times, and now I'm gonna have two twos, so times two times two,
which is equal to four. And you could keep going like that. Now, the reason why I have this one here, and we've done this before, is to justify, and there's many other
good reasons why two 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's about powers of zero, but I've been doing powers of two. So let's focus on zero now. So 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
one times zero twice, so times zero times zero. 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 one times zero that positive integer number of times. So once again, it's going
to be equal to zero. And in general, you can extend that. Zero to any positive value exponent is 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, the zero to the zeroth power
should be equal to one. Other folks would say, "Hey, no, I'm with the zero and
that's the zeroth power, maybe it should be a zero." And that's why a lot of
folks leave it undefined. Most of the time, you're going to see zero
to the zeroth power, either being undefined or
that it is equal to one.