WEBVTT
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This is part four of rational exponents, and we're just going to work through more problems.
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So first, just a reminder about some of the laws of exponents we've used, along with
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the new ones on rational expressions.
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All right, so we're going to use the properties of exponents to simplify each expression,
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and only write with positive exponents.
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So here's one problem.
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27 u cubed to the 2 thirds.
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See if you could try this on your own first.
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All right, let's see.
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So what we have to do is raise each thing inside the parentheses to the 2 thirds power.
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So we have 27 to the 2 thirds times u cubed to the 2 thirds.
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Now what's 27 to the 2 thirds mean?
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It means the cube root of 27 squared.
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So the cube root of 27 is 3, and then you could square it.
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Now, some of you can do that all in one step.
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I'm just doing the cube root of 27 in my head to write the 3.
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But over here, we're going to use our law of exponents, multiply in exponents.
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Well, when you multiply 3 times 2 thirds, the 3's cancel, so you get u squared.
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So then we get 9 u squared for our answer.
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All right, so that's first problem.
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All right, how about x to the 1 fourth times x to the negative 1 half over x to the 2 thirds?
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Well, all of these have the same base x, so what you can do is add the exponents in the
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numerator and subtract the one in the denominator.
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You can do this in two steps or one step.
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I'm just going to write.
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This is going to be 1 fourth.
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Then I'm going to add a negative 1 half.
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So I'm going to write minus 1 half, and then plus 2 thirds.
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And I'm just going to simplify the fractions down here.
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So I've got 1 fourth minus 1 half plus 2 thirds.
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You can either just do the 1 fourth minus 1 half first, or you can just get a common denominator
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for all of them, which is what I would suggest.
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So for 4, 2, and 3, what's the common denominator?
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It's going to be 12.
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So we want to write those all with the denominator of 12.
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So for 1 fourth, we're going to multiply by 3 over 3.
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So that's going to be 3 thirds.
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For 1 half, we're going to multiply the numerator and denominator by 6 over 6.
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So that's 6, 12.
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Number 2 third, we're going to multiply the numerator and denominator by 4.
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So remember that that's not just a 1 on the top.
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When you do that 4 over 4, you're going to get a 8 in the numerator, right?
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Okay, so then be careful of your subtraction.
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You're doing 3 minus 6, that's a negative 3 plus 8.
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So that's going to be 5, 12.
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So this is going to be x to the 5, 12.
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To be honest, most people have more trouble with adding and subtracting fractions and using
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the laws of exponents.
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So this is correct answer.
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You've written it with a positive exponent.
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If you want, you can write it as the 12th root of x to the 5th as well.
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That's also correct, that's just writing it in radical form.
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All right, this is a complicated problem and there are a lot of different ways to do
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this problem.
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There isn't just one way you might go about doing it.
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What I'm going to do is go ahead and just raise each thing in each parenthesis to the
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exponent.
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So that's one way you can start.
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So I'm going to have a to the negative 2 to the 1-8th, right?
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And b cubed to the 1-8th.
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And in the denominator, I'm going to do the same thing.
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I'm going to get a to the negative 3 to the negative 1-4th and b to the negative 1-4th.
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So let's simplify by each of the exponents by multiplying using the law of exponents
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where you multiply these two exponents.
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So I have to do negative 2 times 1-8th.
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That's going to give you an a.
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Now it's negative 2-8th.
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You could leave it like that or you could reduce it to negative 1-4th.
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So right now I'm just going to leave it as negative 2-8th.
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And I'm going to have a b 3 times 1-8th.
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That'll be 3-8th.
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Okay.
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Let's see.
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The denominator, if I multiply that gives me a to the negative 3 times the negative 1-4th
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is a to the 3-4th.
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And b to the negative 1-4th.
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Okay.
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Now as it turns out, I could sort of see my green of a's here together.
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It would be easier if I did change that to negative 1-4th.
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But also since it's a negative exponent, I'm going to put that in the denominator.
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So watch what happens.
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I'm going to circle the ones I'm going to put in the denominator.
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Anything that has a negative exponent, it switches from numerator to denominator or vice
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versa.
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So if I write that in the denominator, it would be a plus 2-8th.
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I'm just going to write that as a to the positive 1-4th.
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And I also have a to the 3-4th.
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It's already down there.
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Okay.
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So I've taken care of the 2 a's.
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Now let's deal with b's.
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The b to the 3-8th stays up there.
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And this b to the negative 1-4th, that goes in the numerator.
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But I'm going to think ahead.
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I'm going to have to add it to this 3-8th because I've got the same base.
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So I'm going to write that instead of positive 1-4th, I'm going to write that as 2-8th.
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I'm just going to write it with a common denominator.
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So now, let's just do the last part.
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We're going to add b to 3-8th times b to the 2-8th.
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We're going to add the exponent.
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So that's b to the 5-8th.
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Whoops.
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Look at that twice.
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b to the 5-8th.
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And in the denominator, what happens here?
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I got a to the 1-4th times a to the 3-4th.
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When I add that, 1-4th plus 3-4th is 1.
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So that's a to the first power, which is just a.
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So that's perfectly fine to just write your answer like that.
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And if you want, you could write b to the 5-8th in radical form by writing the 8-3 to b to
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the 5th.
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So, but this is perfectly acceptable the way I've done it.
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Okay.
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We're going to work on some different kind of problems.
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I'll see all variables represent positive or real numbers.
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And then we're going to use rational exponents to simplify each radical.
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So this means now I want to change from radical notation to rational to see if that helps.
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All right.
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So we've got 8 cubed to the, well, the 18-3.
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That just means to the 1-18th.
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And then we can multiply exponents.
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So that will give you 3-18-3-1-6.
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Now we can go back and rewrite it.
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So that's the same thing as just the 6-3.
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Okay.
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Let's look at a different one.
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How about the 4-3 to 36?
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Okay.
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There's a little bit tricky.
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What you want to notice here is if something is a perfect square or a perfect cube, you want
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to write it in that form.
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And 36 is 6 squared.
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So watch this.
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Instead of writing 36, I'm going to write 6 squared.
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And that's 2-the-1-4th power because it's the 4-3.
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And now we're going to multiply the exponents.
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So I get 6 to the 2 times 1-4 is 1-half.
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And that means the square root is 6.
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So you can see there's an easier way of writing the 4-3 to 36.
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It's just the same thing as the square root of 6.
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All right.
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Let's do another one.
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We've got the 8-3 to have x plus 2 to the 4th power.
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All right.
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So we have x plus 2 to the 4th power.
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And then that, whole thing is, since it was the 8-3, that is to the 1-8th power.
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So when we multiply the exponents, that's 1-half.
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And what's the easiest way of writing x plus 2 to the 1-half?
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We're root of x plus 2.
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Here's another one.
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The 12th root of 8-9th, b to the 8th.
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OK.
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So we start off with 8-9th, b to the 8th to the 1-12.
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And so we have 8-9th to the 1-12th power times b to the 8th to the 1-12th power.
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So what's that give me?
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8 to the 1-9th to the 9-12th that reduces to 3-4th and b 8-12th that reduces to 2-3rd.
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So we could write this as 8 to the 3-4th, b to the 2-3rd, or you could put that back into
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radical, which would be the 4th root of a cubed times the cube root of b squared.
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So the these are correct.
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So this one is in exponential form and what is in radical form.