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In this video, I'm going to do some more examples of

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simplifying radical expressions.

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But these are going to involve adding and subtracting

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different radical expressions.

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And I think it's a good tool to have in your toolkit in

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case you've never seen it before.

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So let's do a few of these.

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So let's say I have 3 times the square root of 8-- we

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learned before that's actually the principal square root of

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8, or the positive square root of 8-- minus 6 times the

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principal square root of 32.

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So let's see what we can do to simplify this.

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So first of all, 8, we can write that as 2 times 4.

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When 4 is a perfect square, you might

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already recognize that.

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We could further factor that into 2 times 2.

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But I don't think we need to.

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So we can rewrite 3 square root of 8 as 3 times the

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square root of 4 times the square root of 2.

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This is the same thing as the square root of 4 times 2,

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which is the square root of 8.

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So this term is the same thing as that term.

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And then, let's look at 32.

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We want to do the square root of 32.

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32 is 2 times 16.

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Once again, 16's a perfect square, so

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we could stop there.

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If you didn't realize that, you would factor

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that as 4 times 4.

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You'd see that twice.

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You could even go even further down to 2 times 2 and all of

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that, but you see immediately that's a perfect square, so we

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can stop there.

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So this second expression can be written as minus 6 times

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the square root of 16 times the square root of 2.

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This right here-- I want to be clear-- is the same thing as

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the square root of 16 times 2.

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You can separate out.

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The square root of 16 times 2 is the square root of 16 times

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the square root of 2.

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We saw that with our exponent properties.

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Now, what does this first term simplify to?

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This is 3 clearly.

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This right here is a 2.

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So you have 3 times 2 times the square root of 2.

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That is 6 times the principal root of 2.

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And then from that we're going to subtract-- well, what's

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this term right here?

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That is positive 4.

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So 6 times 4 is 24 times the square root of 2.

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And we're not done yet.

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If I have 6 of something and I'm going to subtract from

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that 24 of that same something, what do I have?

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I have 6 square roots of 2 and I'm going to subtract from

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that 24 square roots of 2, well, this is going to be

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equal to 6 minus 24 is negative 18 square roots of 2.

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And hopefully, this doesn't confuse you.

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Remember, if we had 6x minus 24x, we would have minus 18x

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or negative 18x.

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Now, instead of an x, we just have a square root of 2.

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6 of something minus 24 of something will get us negative

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18 of that something.

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Let's do another one.

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Let's say I have the square root of 180 plus 6 times the

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square root of 405.

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So this is really an exercise in being able to simplify

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these radicals, which we've done before.

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But you can never get too much practice doing that.

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So let's just do the factorization

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of these right here.

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So 180 is 2 times 90, which is 2 times 45,

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which is 5 times 9.

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And we can factor 9 down more into 3 times 3 to realize it's

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a perfect square, but we could leave it like that.

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So this first term right here we can write as the square

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root of 2 times 2 times the square root of 5 times the

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square root of 9.

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I'm going to put the square root of 9 out front.

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So square root of 2 times 2 times the square root of 5

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times the square root of 9.

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Now, what is this second term equal to?

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So let's factor it out.

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405.

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That is 5 times-- I think it's 81.

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But just to verify, 405, 5 doesn't go into 4, so

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let's go into 40.

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5 goes into 40 eight times.

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8 times 5 is 40.

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Subtract.

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You get a 0.

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Bring down the 5.

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5 goes into 5 one time.

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Right, 81 times.

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81 is 9 times 9.

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You could factor more if we were trying to do the fourth

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root or something like that, but we want to just do a

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square root.

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We have a 9 and a 9, so no need to factor any more.

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So this second expression right here is plus 6 times the

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square root of 9 times 9 times the square root of 5.

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So what is this?

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This is 3.

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This is 2.

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This is the square root of 4.

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So it's 3 times 2 is 6.

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So we have 6 square roots of 5.

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Plus-- what's this right here?

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The square root of 9 times 9, the square root of 81.

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That's, of course, just 9.

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So 6 times 9 is 54, so plus 54 square roots of 5.

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And then, what do we have left?

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We have 6 of something plus 54 of something.

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That's going to be equal to 60 of that

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something just like that.

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Let's just do one more and we're going to have some

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abstract quantities here.

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We're going to deal with some variables.

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But I really just want to do it to show you that the

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variables don't change anything.

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Let's say if we have the square root or the principal

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root of 48a.

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And I'm going to add that to the square root of 27a.

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So once again, let's just factor the 48 part.

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We'll leave the a aside.

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So 48 is 2 times 24, which is 2 times 12.

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Sorry, 2 times 12, which is 3 times 4.

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So we could rewrite this first expression here as the square

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root of 2 times 2 times the square root of 4 times the

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square root of 3.

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Now, you might have done it a quicker way.

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You might have just factored into 3 and 16 and immediately

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realized that 16 is a perfect square.

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But I did it just kind of the brute force way.

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You'd get the same answer either way.

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And, of course, not just the square root of 3, you also

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have the square root of a there.

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So I'll just put the a right over here.

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I could put it in a separate square root, but both of these

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aren't perfect squares, so I'll leave both of these under

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the radical sign.

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Now, 27 is 3 times 9.

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9 is a perfect square root, so we can stop there.

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So this second term, we can rewrite it as the square root

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of 9 times the square root of 3a.

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And in both of these you can kind of view it I'm skipping

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an intermediate step.

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The intermediate step, I could have written that first

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expression as the square root of 9 times 3a and then

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gone to this step.

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But I think we have enough practice realizing that 9

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times 3a, all of that to the 1/2 power, or taking the

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principal root of all of that is the same thing as taking

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the principal root of 9 times the principal root of 3a.

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So that's the step I skipped in both of these.

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But hopefully, that doesn't confuse you too much.

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And so, this term right here is going to be a 2.

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This term right here is going to be a 2.

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So this is going to be 4 times the square root of 3a.

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And then this over here, this right here, is a 3.

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So this is going to be plus 3 times the square root of 3a.

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4 of something plus 3 of something will be equal to 7

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of the something.

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Anyway, hopefully, you found that useful.