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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 1 over a to the b.

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And before I give you the intuition, I want you to

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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, and they defined this for the reasons

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that I'm going to show you.

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

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

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you learned exponent rules, all of the other exponent rules

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stay consistent for negative exponents and when you raise

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something to the zero 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 1, a squared,

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a cubed, a to the fourth.

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What's a to the 1? a to the 1, we said, is a, and then to get

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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

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decrease the exponent, what are we doing?

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We are multiplying by 1/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,

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you're dividing by a.

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So let's use this progression to figure out what

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a to the 0 is.

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So this is the first hard one.

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So a to the 0.

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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 0 is.

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And, you know, maybe it's 17, 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 0 is.

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But wouldn't it be nice if a to the 0 retained this pattern?

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That every time you decrease the exponent, you're

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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, and

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divide by a, right, so we're just going to go-- we're just

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going to divide it by a, what is a divided by a?

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Well, it's just 1.

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So that's where the definition-- or that's one of

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the intuitions behind why something to the 0-th

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power is equal to 1.

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Because when you take that number and you divide it by

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itself one more time, you just get 1.

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So that's pretty reasonable, but now let's go into

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the negative domain.

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So what should a to the negative 1 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

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dividing by a.

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So let's divide by a again, so 1/a.

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So we're going to take a to the 0 and divide it by a.

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a to the 0 is one, so what's 1 divided by a?

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It's 1/a.

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Now, let's do it one more time, and then I think you're

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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 2?

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Well, we want-- you know, it'd be silly now to

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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 1 to a to the minus 2, let's

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just divide by a again.

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And what do we get?

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If you take 1/2 and divide by a, you get 1 over a squared.

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

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the left, and you would get a to the minus b is equal

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to 1 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 0-th power, why does that equal 1?

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

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Someone decided it should be equal to 1, but they

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had a good reason.

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And their good reason was they wanted to keep

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this pattern going.

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And that's the same reason why they defined negative

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exponents in this way.

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And what's extra cool about it is not only does it retain this

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pattern of when you decrease exponents, you're dividing by

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

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by a, but as you'll see in the exponent rules videos, all

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of the exponent rules hold.

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All of the exponent rules are consistent with this definition

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of something to the 0-th power and this definition of

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something to the negative power.

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Hopefully, that didn't confuse you and gave you a little bit

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of intuition and demystified something that, frankly,

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is quite mystifying the first time you learn it.

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