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# Powers of zero

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.