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

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a to the minus b is equal to one over a to the b.

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And before I give you the intuition,

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I want you to just realize that this really is a definition.

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I don't know.

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The inventor of mathematics wasn't one person.

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It was, you know, a convention that arose.

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But they defined this,

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and they defined this for the reasons that I'm going to show you.

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Well, what I'm going to show you is one of the reasons,

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and then we'll see that this is a good definition,

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because once you learn the exponent rules, all of the other exponent rules stay consistent for negative exponents

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and when you raise something to the zeroth power.

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So let's take the positive exponents.

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Those are pretty intuitive, I think.

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

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a cubed, a to the fourth.

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What's a to the one? a to the one, we said, was a,

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and then to get to a squared, what did we do?

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We multiplied by a, right?

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a squared is just a times a.

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And then to get to a cubed, what did we do?

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We multiplied by a again.

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And then to get to a to the fourth, what did we do?

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We multiplied by a again.

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Or the other way, you could imagine, is when you decrease the exponent,[br]what are we doing?

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We are multiplying by one over a, or dividing by a.

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And similarly, you decrease again, you're dividing by a.

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And to go from a squared to a to the first, you're dividing by a.

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So let's use this progression to figure out what a to the zero is.

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So this is the first hard one.

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So a to the zero.

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

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

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

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I don't know.

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

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But wouldn't it be nice if a to the zero retained this pattern?

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That every time you decrease the exponent, you're dividing by a, right?

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So, if you're going from a to the first to a to the zero,

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wouldn't it be nice if we just divided by a?

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So let's do that.

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So if we go from a to the first, which is just a,

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and divide by a,

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right, so we're just going to go-- we're just going to divide it by a.[br]

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What is a divided by a?

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Well, it's just one.

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So that's where the definition--

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

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Because when you take that number

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

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So that's pretty reasonable,

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but now let's go into the negative domain.

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So what should a to the negative one equal?

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Well, once again, it's nice if we can retain this pattern,

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where every time we decrease the exponent we're dividing by a.

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So let's divide by a again, so one over a.

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

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

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It's one over a.

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Now, let's do it one more time,

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and then I think you're going to get the pattern.

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Well, I think you probably already got the pattern.

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What's a to the minus two?

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Well, we want-- you know, it'd be silly now to change this pattern.

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Every time we decrease the exponent, we're dividing by a.

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So to go from a to the minus one to a to the minus two,

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let's just divide by a again.

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And what do we get?

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If you take one over a and divide by a, you get one over a squared.

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And you could just keep doing this pattern all the way to the left,

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and you would get a to the minus b is equal to one over a to the b.

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Hopefully, that gave you a little intuition as to why--

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well, first of all, you know, the big mystery is, you know,

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something to the zeroth power, why does that equal one?

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

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Someone decided that it should be equal to one, but they had a good reason.

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And their good reason was they wanted to keep this pattern going.

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And that's the same reason why they defined negative exponents in this way.

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And what's extra cool about it is

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not only does it retain this pattern of when you decrease exponents, you're dividing by a,

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or when you're increasing exponents, you're multiplying by a,

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but as you'll see in the exponent rules videos, all of the exponent rules hold.

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All of the exponent rules are consistent with this definition of something to the zeroth power

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and this definition of something to the negative power.

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Hopefully, that didn't confuse you

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and gave you a little bit of intuition and demystified something that, frankly,

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is quite mystifying the first time you learn it.