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# Powers of zero

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.