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So a few videos ago, I told you that anything to the

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zeroth power is equal to 1.

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So x to the zeroth power is equal to 1.

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And I gave you one argument why this is the case.

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I used the example of, if we have 3 to the first power,

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that is equal to 3.

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3 to the second power is equal to 9.

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3 to the third power is equal to 27.

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So every time we decrease by a power, we're dividing by 3.

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27 divided by 3 is 9.

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9 divided by 3 is 3.

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Then 3 divided by 3 is 1.

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And that should be what 3 to the zeroth power is.

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So that's one way to think about it.

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The other way to think about it is that we need this for

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the exponent properties to work.

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For example, I told you that a to the b times a to the c is

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equal to a to the b plus c.

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Now, what happens if c is 0?

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What happens if we have a to the b times a to the 0?

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Well, by this property, this needs to be equal to a to the

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b plus 0, which is equal to a to the b.

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So a to the b times a to the 0 must be equal to a to the b.

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If you divide both sides of this times a-- let me rewrite

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this-- a to the b times a to the 0, if we use this property

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up here, must be equal to a to the b, right? b plus 0 is b.

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If you divide both sides by a to the b, what do you get?

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On the left-hand side, you're left with

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just a to the 0, right?

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These cancel out.

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a to the 0 is equal to 1.

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And you can use a similar argument in pretty much all of

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the exponent properties, that we need anything to the zeroth

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power to be equal to 1.

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And it also makes sense as we divide by 3, each step as we

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decrement our exponent.

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It keeps working.

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When you take 3 to the negative 1 power, we saw on

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the last video that that's equal to 1 over 3 to the first

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power, or 1/3 .

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So once again, from 3 to the 0 to 1/3, you're

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dividing by 3 again.

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So it really makes sense on some level that 3 to the

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zeroth power is equal to 1.

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But that leaves a little bit of a gap.

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What about 0 to the zeroth power?

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This is a very strange notion.

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0 multiplied by itself 0 times.

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And it depends what context you're using.

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Sometimes people will say that this is undefined, but many

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more times, at least in my experience, this'll be

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defined to be 1.

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And the reason why-- even though this is completely not

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intuitive, and you could type in 0 to the zeroth power in

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Google, and it'll give you 1.

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Even though this is completely not intuitive, the reason why

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this is defined to be this way is that makes a lot of

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formulas work.

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One in particular, the binomial formula works for

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your binomial coefficients, which I'm not going to go over

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right here, when 0 to the zeroth power is equal to 1.

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So that's an interesting thing for you to think about, what

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that might even mean.

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So let's talk about some of the other properties.

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And then we can put them all together with a couple of

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example problems. I told you in the last video what it

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means to raise to a negative power.

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a to the negative 1 power, or maybe I should say a to the

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negative b power is equal to 1 over a to the b power.

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So just to do that with a couple of concrete examples, 3

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to the negative 3 power is equal to 1 over 3 to the third

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power, which is equal to 1 over 3 times 3, times 3, which

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is equal to 1 over 27.

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If I were to ask you what 1/3 to the negative 2 power is--

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well, this is going to be equal to 1 over 1/3 to the

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second power.

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You get rid of the negative and you inverse it.

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So this is going to be equal to 1 over--

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what's 1/3 times 1/3?

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1/9.

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Which is equal to-- this is 1 divided by 1/9 is the same

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thing is 1 times 9, so this is equal to 9.

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And this makes complete sense, because 1/3, remember, 1/3 is

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the same thing as 3 to the negative 1 power, right?

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3 to the negative 1 is equal to 1 over 3 to the 1 power,

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which is the same thing is 1/3.

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So if we replace 1/3 with 3 to the negative 1, this is 3 to

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the negative 1 to the negative 2 power.

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These two things are equivalent statements.

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And if we use one of the properties we learned in the

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first video, we can take the product

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of these two exponents.

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So this is equal to 3 to the negative 1, times negative 2,

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which is just positive 2, which is equal to 9.

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So it's really neat how all of these exponent properties

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really fit together in a nice, neat puzzle, that they don't

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contradict each other.

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And it doesn't matter which property you use, you'll get

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the right answer in the end, as long as you don't do

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something crazy.

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Now, the last thing I want to define is the notion of a

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fractional exponent.

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So if I have something to a fractional power-- so let's

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say I have a to the 1 over b power.

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I'm going to define this.

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This is going to be equal to the bth root of a.

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So let me be very clear here.

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Let me make it with some numbers here.

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If I said 4 to the 1/2 power right there, this means this

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is equivalent to the square root of 4.

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Which is equal to, if we're taking the principal root,

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this is equal to 2.

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So if I were to take, let's be clear, 8 to the 1/3 power,

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this is taking the cube root of 8.

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And this is, on some level, one of the most sometimes

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confusing things in exponents.

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Here I'm saying, what number times itself 3 times

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is equal to 8 ?

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So if I said that x is equal to 8 to the 1/3 power, this is

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the exact same thing as saying x to the third

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power is equal to 8.

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And how do I know that these are equivalent statements?

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Well, I could take both sides of this equation

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to the third power.

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If I take the left-hand side of the third power and the

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right-hand side of the third power, what do I get?

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On the left-hand side, I get x to the third.

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On the right-hand side, I get 8 to the 1/3 times 3, which is

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just 3 over 3, which is just 1.

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So if x is equal to 8 to the 1/3, what is x?

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Well, 2 times 2, times 2 is equal to 8.

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And there's no really easy way, especially once you go to

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the fourth root, or the fifth root, and you have decimals of

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calculating these.

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You probably need a calculator most of the time to do these.

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But things like 8 to the 1/3, or 16 to the 1/4, or 27 to the

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1/3 , they're not too hard to calculate.

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So this right here, let me be clear, is 2.

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Now, let's make it a little bit more confusing.

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What is 27 to the negative 1/3 power?

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Well, don't get too worried.

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We're just going to take it step by step.

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When you take the negative power, this is completely

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equivalent to 1 over 27 to the 1/3 power.

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These two are equivalent.

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You get rid of the negative and take 1

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over the whole thing.

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And then what is 27 to the 1/3 power?

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Well, what number times itself 3 times is equal to 27?

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Well, that's equal to 3.

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So this is going to be equal to 1 over 3.

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Not too bad.

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Now I'm going to take it even to another level, make it even

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more confusing, even more daunting.

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Now, let me do something interesting.

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What is 8 to the 2/3 power?

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Now that seems a little bit scary.

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And all you have to remember is this is the same thing,

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using our exponent rules really, as 8

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squared to the 1/3 power.

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How do I know that?

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Well, if I multiply these two exponents, this is 2/3.

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So 8 to the 2/3 is the same thing is 8 squared, and then

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the third root of that.

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But you could view it the other way.

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This should also be equal to 8 to the 1/3 power squared.

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Because either way, when I multiply these exponents, I

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get 8 to the 2/3.

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Let's verify for ourselves that we really do

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get the same value.

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So 8 squared is 64.

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And we're going to take that to the 1/3 power.

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Down here, we have 8 to the 1/3.

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We already figured out what that is.

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That's 2, because 2 to the third power is 8.

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So this is 2 squared.

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Now, what is 64 to the 1/3?

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What times itself 3 times is equal to 64?

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Well, 4 times 4, times 4 is equal to 64, or 4 to the third

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is equal to 64, which means that 4 is equal

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to 64 to the 1/3.

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So this is equal to 4.

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And, lucky for us, 2 squared is also equal to 4.

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So it doesn't matter which way you do it.

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You could take the square and then the third root, or you

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could take the third root and then square it.

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You're going to get the exact same answer.

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Now, everything I've been doing has

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been with actual numbers.

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Let me do a couple of problems that just bring everything

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we've done together using variables.

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So let's say we wanted to do a few expressions and we want to

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make sure there are no negative

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exponents in the answer.

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So let's add x to the negative 3 over x to the negative 7.

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There's a bunch of ways we could view this.

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We could view this as equal to x to the negative 3, times 1

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over x to the negative 7.

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And what is 1 over x to the negative 7?

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This is the same thing as x to the seventh power, right?

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If you have 1 over something, you can get rid of the 1 over,

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and put a negative in front of the exponent.

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But if you're putting a negative in front of a

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negative 7, you're going to get x to a seventh.

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So this thing can simplify to x to the negative 3, times x

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to the seventh power.

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And then we can add the exponents, and that is x to

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the fourth power.

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Now, another way, a completely legitimate way we could have

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done this, is we could have just subtracted the exponents.

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We could have said, well, gee, this is the same base.

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This is going to be x to the negative 3, minus negative

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seventh power.

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Well, negative 3 minus negative 7, that's a negative

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3 plus 7 which is equal to x to the fourth power.

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And then one final way-- I mean, actually, there's more

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than one final way we could have done this.

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We could have said x to the negative 3 over x to the

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negative 7-- sorry, not negative x-- over x to the

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negative 7.

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Well, x to the negative 3 is the same thing as 1 over x to

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the third-- that's that term right there-- times 1 over x

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to the negative 7, so this would have been equal to 1

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over x to the third times x to the negative 7.

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You could add the exponents, so that's equal to 1 over 3

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minus 7 is x to the negative 4.

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And then this-- if we just get rid of the inverse, we take

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the inverse of it, we can put a negative in front of this

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negative, making it a positive-- this is going to be

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equal to x to the 4.

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So no matter how we did it, as long as we're consistent with

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the rules, we got x to the fourth.

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Let's do one more slightly hairy one.

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And then I think we'll be done for now.

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Let's say we have 3x squared times y to the 3/2 power.

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And we're going to divide it by x times y to the 1/2 power.

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Well, once again, this is the same thing as 3 times the x

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terms right here, so 3 times x squared over x, times y to the

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3/2 over y to the 1/2.

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Well, this is going to be equal to 3 times-- what's x

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squared over x?

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Or x squared over x to the first power?

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That's going to be equal to x to the 2 minus 1.

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And then this is going to be times y to the 3/2 minus 1/2.

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So what does the whole thing become?

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It becomes 3 times x.

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2 minus 1 is just 1-- I can just write x there-- times 3/2

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minus 1/2 is 2/2.

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So that's y to the 2/2.

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2/2, or 2 2ths-- that's just the same thing is y.

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so this is equal to 3xy.

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Anyway, I encourage you to do many, many

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more examples of that.

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But, you'll see that just using the rules that we've

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been exposed to in the last few videos, you can pretty

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much simplify any exponent expression.

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