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## Exponents with negative bases

## Video transcript

What I want to do
in this video is think about exponents in a
slightly different way that will be useful for
different contexts and also go through
a lot more examples. So in the last video, we
saw that taking something to an exponent means multiplying
that number that many times. So if I had the
number negative 2 and I want to raise
it to the third power, this literally means
taking three negative 2's, so negative 2, negative
2, and negative 2, and then multiplying them. So what's this going to be? Well, let's see. Negative 2 times
negative 2 is positive 4, and then positive 4 times
negative 2 is negative 8. So this would be
equal to negative 8. Now, another way of
thinking about exponents, instead of saying you're just
taking three negative 2's and multiplying them, and this
is a completely reasonable way of viewing it, you
could also view it as this is a number
of times you're going to multiply
this number times 1. So you could
completely view this as being equal to-- so you're
going to start with a 1, and you're going to multiply 1
times negative 2 three times. So this is times negative
2 times negative 2 times negative 2. So clearly these
are the same number. Here we just took this, and
we're just multiplying it by 1, so you're still going
to get negative 8. And this might be a
slightly more useful idea to get an intuition for
exponents, especially when you start taking things
to the 1 or 0 power. So let's think about
that a little bit. What is positive
2 to the-- based on this definition-- to the
0 power going to be equal to? Well, we just said. This says how many times are
going to multiply 1 times this number? So this literally says,
I'm going to take a 1, and I'm going to
multiply by 2 zero times. Well, if I want to multiply
it by 2 zero times, that means I'm just
left with the 1. So 2 to the zero power is
going to be equal to 1. And, actually, any non-zero
number to the 0 power is 1 by that same rationale. And I'll make another
video that will also give a little bit more
intuition on there. That might seem very
counterintuitive, but it's based on
one way of thinking about it is thinking
of an exponent as this. And this will also
make sense if we start thinking of what
2 to the first power is. So let's go to this definition
we just gave of the exponent. We always start with a 1, and we
multiply it by the 2 one time. So 2 is going to
be 1-- we're only going to multiply it by the 2. I'll use this for
multiplication. I'll use the dot. We're only going to
multiply it by 2 one time. So 1 times 2, well,
that's clearly just going to be equal to 2. And any number to
the first power is just going to be
equal to that number. And then we can go from there,
and you will, of course, see the pattern. If we say what 2 squared is,
well, based on this definition, we start with a 1, and we
multiply it by 2 two times. So times 2 times 2 is
going to be equal to 4. And we've seen this before. You go to 2 to the third,
you start with the 1, and then multiply
it by 2 three times. So times 2 times 2 times 2. This is going to
give us positive 8. And you probably
see a pattern here. Every time we multiply by 2--
or every time, I should say, we raise 2 to one more power,
we are multiplying by 2. Notice this, to go from
2 to the 0 to 2 to the 1, we multiplied by 2. I'll use a little x for the
multiplication symbol now, a little cross. And then to go from 2
to the first power to 2 to the second power, we multiply
by 2 and multiply by 2 again. And that makes complete sense
because this is literally telling us how many times are
we going to take this number and-- how many times are we
going take 1 and multiply it by this number? And so when you go from 2 to the
second power to 2 to the third, you're multiplying
by 2 one more time. And this is another intuition
of why something to the 0 power is equal to 1. If you were to go
backwards, if, say, we didn't know what
2 to the 0 power is and we were just
trying to figure out what would make
sense, well, when we go from 2 to the third
power to 2 to the second, we'd be dividing by 2. We're going from 9 to 4. Then we'd divide by 2 again to
go from 2 to the second to 2 to the first. And then it seems
like we should just divide by 2 again from
going from 2 to the first to 2 to the 0. And that would give us 1.