Negative Exponent Intuition

Intuition on why a^-b = 1/(a^b) (and why a^0 =1)

I have been asked for some intuition as to why, let's say,

a to the minus b is equal to one 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 learn the exponent rules, all of the other exponent rules stay consistent for negative exponents

and when you raise something to the zeroth power.

So let's take the positive exponents.

Those are pretty intuitive, I think.

So the positive exponents, so you have a to the one, a squared,

a cubed, a to the fourth.

What's a to the one? a to the one, we said, was 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 one over 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 zero is.

So this is the first hard one.

So a to the zero.

So you're the inventor, the founding mother of mathematics,

and you need to define what a to the zero is.

And, you know, maybe it's seventeen, maybe it's pi.

I don't know.

It's up to you to decide what a to the zero is.

But wouldn't it be nice if a to the zero 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 one.

So that's where the definition--

or that's one of the intuitions behind why something to the zeroth power is equal to one.

Because when you take that number

and divide it by itself one more time, you just get one.

So that's pretty reasonable,

but now let's go into the negative domain.

So what should a to the negative one 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 one over a.

So we're going to take a to the zero and divide it by a.

a to the zero is one, so what's one divided by a?

It's one over 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 two?

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 one to a to the minus two,

let's just divide by a again.

And what do we get?

If you take one over a and divide by a, you get one 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 one 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 zeroth power, why does that equal one?

First, keep in mind that that's just a definition.

Someone decided that it should be equal to one, 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 zeroth 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.