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# Negative exponent intuition

Video transcript

I have been asked for some
intuition as to why, let's say, a to the minus b is equal
to 1 over a to the b. And before I give you the
intuition, I want you to just realize that this
really is a definition. I don't know. The inventor of mathematics
wasn't one person. It was, you know, a
convention that arose. But they defined this, and they
defined this for the reasons that I'm going to show you. Well, what I'm going to show
you is one of the reasons, and then we'll see that this is a
good definition, because once you learned exponent rules, all
of the other exponent rules stay consistent for negative
exponents and when you raise something to the zero power. So let's take the
positive exponents. Those are pretty
intuitive, I think. So the positive exponents, so
you have a to the 1, a squared, a cubed, a to the fourth. What's a to the 1? a to the 1,
we said, is a, and then to get to a squared, what did we do? We multiplied by a, right? a squared is just a times a. And then to get to a
cubed, what did we do? We multiplied by a again. And then to get to a to the
fourth, what did we do? We multiplied by a again. Or the other way, you could
imagine, is when you decrease the exponent,
what are we doing? We are multiplying by
1/a, or dividing by a. And similarly, you decrease
again, you're dividing by a. And to go from a squared
to a to the first, you're dividing by a. So let's use this progression
to figure out what a to the 0 is. So this is the first hard one. So a to the 0. So you're the inventor, the
founding mother of mathematics, and you need to define
what a to the 0 is. And, you know, maybe
it's 17, maybe it's pi. I don't know. It's up to you to decide
what a to the 0 is. But wouldn't it be nice if a to
the 0 retained this pattern? That every time you decrease
the exponent, you're dividing by a, right? So if you're going from a to
the first to a to the zero, wouldn't it be nice if
we just divided by a? So let's do that. So if we go from a to the
first, which is just a, and divide by a, right, so we're
just going to go-- we're just going to divide it by a,
what is a divided by a? Well, it's just 1. So that's where the
definition-- or that's one of the intuitions behind why
something to the 0-th power is equal to 1. Because when you take that
number and you divide it by itself one more time,
you just get 1. So that's pretty reasonable,
but now let's go into the negative domain. So what should a to
the negative 1 equal? Well, once again, it's nice if
we can retain this pattern, where every time we decrease
the exponent we're dividing by a. So let's divide by
a again, so 1/a. So we're going to take a to
the 0 and divide it by a. a to the 0 is one, so
what's 1 divided by a? It's 1/a. Now, let's do it one more
time, and then I think you're going to get the pattern. Well, I think you probably
already got the pattern. What's a to the minus 2? Well, we want-- you know,
it'd be silly now to change this pattern. Every time we decrease the
exponent, we're dividing by a, so to go from a to the minus 1
to a to the minus 2, let's just divide by a again. And what do we get? If you take 1/2 and divide by
a, you get 1 over a squared. And you could just keep doing
this pattern all the way to the left, and you would get a
to the minus b is equal to 1 over a to the b. Hopefully, that gave you a
little intuition as to why-- well, first of all, you know,
the big mystery is, you know, something to the 0-th power,
why does that equal 1? First, keep in mind that
that's just a definition. Someone decided it should
be equal to 1, but they had a good reason. And their good reason
was they wanted to keep this pattern going. And that's the same reason
why they defined negative exponents in this way. And what's extra cool about it
is not only does it retain this pattern of when you decrease
exponents, you're dividing by a, or when you're increasing
exponents, you're multiplying by a, but as you'll see in the
exponent rules videos, all of the exponent rules hold. All of the exponent rules are
consistent with this definition of something to the 0-th power
and this definition of something to the
negative power. Hopefully, that didn't confuse
you and gave you a little bit of intuition and demystified
something that, frankly, is quite mystifying the
first time you learn it.