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This is part 8 of square roots and radicals and we're going to expand our work with radicals
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by discussing cube roots, 4th roots, etc.
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Let's begin with looking at perfect cubes, perfect 4ths, etc.
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So we have some perfect cubes here in green on the left.
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Two cubed, remember means 2 times 2 times 2 is 8.
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Three cubed, 3 times 3 times 3 is 27, etc.
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So for instance, you can see some of the perfect cubes would be 8, 27, 64, 125, 216,
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343.
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I've memorized up to that but you don't have to memorize that far.
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You could always figure it out by multiplying the three numbers out together again, right?
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5 cubed, you would just do 5 times 5 times 5 and that's how you would get 125.
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If we look at some of the common perfect 4ths, that would be 2 to the 4th power, would be
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16.
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3 to the 4th power is 81.
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Remember you get that by doing 3 times 3 times 3, which is really just 9 times 9, or 81.
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And we have 4 to the 4th, 5 to the 4th, etc.
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Now for unit of 5th power, I only know the first couple of those.
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2 to the 5th is 32.
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3 to the 5th is 243.
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And once we're after the 5th power, I really only know them for 2.
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2 to the 6th, I figure out, is 64.
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You just keep multiplying by 2 and you get up to the 2 to the 10th pretty easily.
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And what I do remember is that 2 to the 10th is approximately 1000.
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It's a little bit bigger than 1000.
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It's a 1000, 24.
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All right.
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So we have to have an understanding of what the perfect cube, the perfect 4th, the perfect
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5th, etc., at least some of them, or be able to figure them out.
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And now when we use a radical sign, a little bit differently, for instance, when we write
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this symbol, we think of this as the square root of 4.
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Right?
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Okay.
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Well, square root of 4 can be written as a little 2 in this little spot here.
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Okay.
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But it's usually left off because we recognize this is the common radical sign, this square
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root without the 2.
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You can write it either way, but basically it still means 2, right?
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Same thing.
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If I want to do the square root of 25, you could write it with a little 2.
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And that tells you this.
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What 2 numbers will apply to it by themselves is 25, and that would be 5.
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Okay.
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So this is called the index, this little number in here.
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So when you have, this is the radical sign.
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It's called the radical sign.
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And the number in here, the little n, is called the index.
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Right?
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So what would this mean?
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Put a little 3 in there, and I put 64.
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What it means is what number cubed equals 64?
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All right.
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So you could either memorize that or you could try to figure it out by say, well, is it 2
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times 2 times 2?
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No, is it 3 times 3 times 3?
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No, is it 4 times 4 times 4?
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And yes, it is.
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So the cube root of 64 is 4.
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What if I did the cube root of 27, I would say what number to the third power is 27, and
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that would be 3.
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What if I wanted to do the fourth root of 81?
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This would mean what number to the fourth power is 81, and that would be 3.
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Either you memorize it, or you could break down 81 to figure that out.
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So if you didn't know what it was, you could say, well, 81, and I'll say that's 9 times
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9, just doing a little factor tree.
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And then you say, oh, 81 really is 3 to the fourth power.
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So the fourth root of 81 is going to be 3.
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All right.
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Try these four problems.
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Put it on pause, then come back.
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All right, let's do the first one.
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Cube root of 125.
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What cubed is 125?
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5.
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Okay.
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What do the fifth power is 32?
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Here's a little hint.
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When we're starting talking about the fifth, sixth power is probably 2.
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So you could check it out and see if it is 2 to the fifth, and in fact it is.
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So 5, 3 to 32 is 2.
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This is a little bit tricky one.
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Fourth root of 10,000.
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Well, the key in here is all those zeros is probably 10 to some power.
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And in fact, this is 10 to the fourth.
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So the fourth root of 10 to the fourth would be 10.
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Squared root of 81, notice I don't put the 2 in there because we don't usually do it
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for square root, but you can, if you'd like, and that would say what square root of 81
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in the answer would be 9.
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And in fact, we have a property we could use.
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And we're going to assume the variables are positive if A is greater than or equal to
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0, since it's not negative then, the nth root of A to the n is just A.
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So if I look at this fourth root of 10,000 here, you could think of this as the fourth
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root of 10 to the fourth would just be 10.
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If I wrote 27 as something cubed, and in this case it happens to be 3, it would just be
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the base here of 3.
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So in this case for A is 3.
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Let's take this a step further and do the same thing we did with simplifying square
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roots that were not perfect squares.
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What we do if we had to do the cube root is something that was not a perfect cubed, like
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40 for instance.
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What we have to do is think of is there a perfect cube that's a factor of 40.
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Over on the right here, I'm going to write my perfect cubes down, at least the first
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few of them, 2 cubed, 3 cubed, 4 cubed, to any of these numbers go into 40 and 8 does.
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So I could rewrite this as the cube root of 8 times 5.
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And now I've written it as a perfect cube times the 5.
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So what's the cube root of 8?
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I could take that out.
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Remember we write that out in front.
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So I have 2 cubed root of 5.
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So that's the answer simplified.
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So a cube root is not simplified if there's a perfect cube that's a factor of 40.
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That's a mouthful, but I don't know any easier way to say that.
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All right, let's do another one.
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Cube root of 54.
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So we want to say there's a perfect cube to factor of 54.
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If you're not sure off the bat, the hint is try dividing by 2 and see if the other number
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is a perfect cube.
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And if not try dividing by 3, et cetera.
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So if we do 54 divided by 2, we get 27.
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And aha, that is a perfect cube.
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So I could rewrite this as the cube root of 27 times 2.
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And now I take out the perfect cube.
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So my answer, the 3 I put out in front and cube root of 2 is left inside.
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And so we've got a answer.
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Now let's look at variables.
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What if I take something like m to the 8th and cube it?
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I get m to the 24th using a rules of exponents, laws of x, when it's going to be multiplied
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8 times 3.
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If you're going to get a multiple of 3, right?
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Because you ended up multiplying this exponent times 3 since you were cubing the whole thing.
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And so to go backwards, if I wanted you to cube root of m to the 24th, basically it's
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going to be a perfect cube to the exponents a multiple of 3.
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For square roots, it had to be even.
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For cubers, it has to be a multiple of 3.
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So since this is a multiple of 3, 24, the exponent is a multiple of 3.
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It's a perfect cube and we just divide by 3 to get back to m to the 8th.
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So let's try another one.
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How about if I wanted to do the cube root of x to the, let's do 18th?
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So first of all, decide is that a perfect cube and why yes it is because it has an exponent
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that 3 goes into.
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So check it out.
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It's almost like you just get to divide this 3 into the 18th.
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That's your new exponent.
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x to the sixth.
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So what if we wanted to do the cube root of m to the 14th?
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No, this 3 doesn't go into 14.
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14 is not a perfect cube.
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So we have to break this down so that we have it as a perfect cube time something.
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So you have to think of the number below 14, the 3 goes into, which will be 12.
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So I have m to the 12th times, using our laws of exponents, we have two extra factors
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of m.
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So that's m to the 12th times m squared.
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And now we could take that out.
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The sum to the fourth, for a final answer, is m to the fourth times the cube root of m squared.
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And we'll do some more problems on the next video.