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## Exponents

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## Video transcript

Let's think about exponents
with ones and zeroes. So let's take the
number 1, and let's raise it to the eighth power. So we've already
seen that there's two ways of thinking about this. You could literally view this
as taking eight 1's, and then multiplying them together. So let's do that. So you have one, two, three,
four, five, six, seven, eight 1's, and then you're
going to multiply them together. And if you were to do that,
you would get well, 1 times 1 is 1, times 1--
it doesn't matter how many times you
multiply 1 by 1. You are going to just get 1. You are just going to get 1. And you could imagine. I did it eight times. I multiplied eight 1's. But even if this was
80, or if this was 800, or if this was 8 million,
if I just multiplied 1-- if I had 8 million 1's, and I
multiplied them all together, it would still be equal to 1. So 1 to any power is just
going to be equal to 1. And you might say, hey,
what about 1 to the 0 power? Well, we've already said
anything to 0 power, except for 0--
that's where we're going to-- it's
actually up for debate. But anything to the 0 power
is going to be equal to 1. And just as a little
bit of intuition here, you could
literally view this as our other definition
of exponentiation, which is you start with a
1, and this number says how many times you're
going to multiply that 1 times this number. So 1 times 1 zero times
is just going to be 1. And that was a little bit
clearer when we did it like this, where
we said 2 to the, let's say, fourth power
is equal to-- this was the other definition
of exponentiation we had, which is you start with a
1, and then you multiply it by 2 four times, so times
2, times 2, times 2, times 2, which is equal to--
let's see, this is equal to 16. So here if you start
with a 1 and then you multiply it by
1 zero times, you're still going to have
that 1 right over there. And that's why anything
that's not 0 to the 1 power is going to be equal to 1. Now let's try some other
interesting scenarios. Let's start try some
negative numbers. So let's take negative 1. And let's first raise
it to the 0 power. So once again,
this is just going, based on this definition,
this is starting with a 1 and then multiplying it
by this number 0 times. Well, that means
we're just not going to multiply it by this number. So you're just going to get a 1. Let's try negative 1. Let's try negative 1
to the first power. Well, anything to the first
power, you could view this-- and I like going with
this definition as opposed to this one right over here. If we were to make
them consistent, if you were to make
this definition consistent with this, you would
say hey, let's start with a 1, and then multiply
it by 1 eight times. And you're still going to
get a 1 right over here. But let's do this
with negative 1. So we're going to
start with a 1, and then we're going to multiply
it by negative 1 one time-- times negative 1. And this is, of course, going
to be equal to negative 1. Now let's take
negative 1, and let's take it to the second power. We often say that
we are squaring it when we take something
to the second power. So negative 1 to the
second power-- well, we could start with a 1. We could start with a 1, and
then multiply it by negative 1 two times-- multiply
it by negative 1 twice. And what's this
going to be equal to? And once again, by
our old definition, you could also just say,
hey, ignoring this one, because that's not going
to change the value, we took two negative 1's
and we're multiplying them. Well, negative 1
times negative 1 is 1. And I think you see
a pattern forming. Let's take negative
1 to the third power. What's this going
to be equal to? Well, by this definition,
you start with a 1, and then you multiply it
by negative 1 three times, so negative 1 times
negative 1 times negative 1. Or you could just
think of it as you're taking three negative 1's
and you're multiplying it, because this 1 doesn't
change the value. And this is going to be equal
to negative 1 times negative 1 is positive 1, times
negative 1 is negative 1. So you see the pattern. Negative 1 to the 0 power is 1. Negative 1 to the first
power is negative 1. Then you multiply
it by negative 1, you're going to get positive 1. Then you multiply it by negative
1 again to get negative 1. And the pattern
you might be seeing is if you take negative
1 to an odd power you're going to get negative 1. And if you take it
to an even power, you're going to get 1 because
a negative times a negative is going to be the positive. And you're going to have an
even number of negatives, so that you're always going to
have negative times negatives. So this right over
here, this is even. Even is going to be positive 1. And then you could see that
if you went to negative 1 to the fourth power. Negative 1 the fourth power? Well, you could start
with a 1 and then multiply it by negative 1 four
times, so a negative 1 times negative 1, times negative
1, times negative 1, which is just going to
be equal to positive 1. So if someone were to
ask you-- we already established that if someone were
to take 1 to the, I don't know, 1 millionth power, this is
just going to be equal to 1. If someone told you
let's take negative 1 and raise it to the 1 millionth
power, well, 1 million is an even number,
so this is still going to be equal to positive 1. But if you took negative
1 to the 999,999th power, this is an odd number. So this is going to be
equal to negative 1.