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Welcome to the presentation on simplifying radicals.

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So let's get started with a little terminology

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out of the way.

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You're probably just wondering what a radical is and

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I'll just let you know.

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A radical is just that.

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Or you're probably more familiar calling that

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the square root symbol.

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So with the terminology out of the way, let's actually talk

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about what it means to simplify a radical.

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And some people would argue that what we're going to

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actually be doing is actually making it more complicated.

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But let's see.

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So if I were to give you the square root of 36.

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Hey, that's easy.

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That's just equal to 6 times 6 or you'd say the square

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root of 36 is just 6.

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Now, what if I asked you what the square root of 72 is?

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Well we know that 72 is 36 times 2.

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Let's write that.

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Square root of 72 is the same thing as the square

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root of 36 times 2.

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And the square root, if you remember from level 3

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exponents, square root is the same thing as something

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to the 1/2 power.

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So let's write it that way.

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And I'm just writing it this way just to show you how this

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radical simplification works and that it's really

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not a new concept.

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So this is the same thing as 36 times 2 to the 1/2 power.

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Because it's just a square root is the same thing as 1/2 power.

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And we learned from the exponent rules that when you

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multiply two numbers and then you raise that to the 1/2

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power, that that's the same thing as raising each of the

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numbers to the 1/2 power and then multiplying.

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Well that right there, that's the same thing as saying the

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square root is 36 times the square root of 2.

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And we already figured out what the square root of 36 is.

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It's 6.

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So that just equals 6 times the square root of 2.

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And you're probably wondering why I went through this step

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of changing the radical, the square root symbol,

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into the 1/2 power.

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And I did that just to show you that this is just an extension

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of the exponent rules.

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It isn't really a new concept, although, I guess sometimes

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it's not so obvious that they are the same concepts.

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I just wanted to point that out.

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So let's do another problem.

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I think as we do more and more problems, these will

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become more obvious.

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The square root of 50.

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Well, the square root of 50 -- 50 is the same

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thing as 25 times 2.

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And we know, based on what we just did and this is really

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just an exponent rule, square root of 25 times 2 is the same

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thing as the square root of 25 times the square root of 2.

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Well we know what the square root of 25 is.

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That's 5.

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So that just equals 5 times the square root of 2.

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Now, you might be saying, "Hey, Sal, you make it look easy, but

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how did you know to split 50 into 25 and 2?" Why didn't I

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say that 50 is equal to the square root of 5 and 10 or that

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50 is equal to the square root -- actually, I think 1 and 50?

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I don't know what other factors is 50.

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Well, anyway, I won't go into that right now.

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The reason why I picked 25 and 2 is because I wanted a factor

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of 50 -- I actually wanted the largest factor of 50 that

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is a perfect square.

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And that's 25.

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If I had done 5 and 10, there's really nothing I could have

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done with it because neither 5 nor 10 are perfect squares and

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same thing's with 1 and 50.

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So the way you should think about it, think about the

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factors of the original number and figure out if any of those

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factors are perfect squares.

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And there's no real mechanical way.

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You really just have to learn to recognize perfect squares.

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And you'll get familiar with them, of course.

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They're 1, 4, 9, 25, 16, 25, 36, 49, 64, et cetera.

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And maybe by doing this module, you'll actually learn to

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recognize them more readily.

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But if any of these numbers are a factor of the number under

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the radical sign, then you'll probably want to

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factor them out.

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And then you can take them out of the radical sign like

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we did up in this problem.

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Let's do a couple more.

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What is 7 times the square root of 27?

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And when I write the 7 right next to it, that just means

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times the square root of 27.

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Well, let's think about what other factors of 27 and

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whether any of them are a perfect square.

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Well, 3 is a factor of 27, but that's not a perfect square.

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9 is.

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So, we could say 7 -- that's equal to 7 times the

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square root of 9 times 3.

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And now, based on the rules we just learned, that's the same

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thing as 7 times the square root of 9 times the

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square root of 3.

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Well that just equals 7 times 3 because the square root of 9 is

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3 times the square root of 3.

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That equals 21 times the square root of 3.

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Done.

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Let's do another one.

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What is 9 times the square root of 18?

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Well once again, what are the factors of 18?

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Well do we have 6 and 3?

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1 and 18?

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None of the numbers I mentioned so far are perfect squares.

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But we also have 2 and 9.

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And 9 is a perfect square.

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Let's write that.

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That's equal to 9 times the square root of 2 times 9.

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Which is equal to 9 times the square root of 2 -- that's a 2

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-- times the square root of 9.

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Which equals 9 times the square root of 2 times 3, right?

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That's the square root of 9 which equals 27 times

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the square root of 2.

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There we go.

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Hopefully, you're starting to get the hang of these problems.

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Let's do another one.

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What is 4 times the square root of 25?

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Well 25 itself is a perfect square.

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This problem is so easy, it's a bit of a trick problem.

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25 itself is a perfect square.

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The square root is 5, so this is just equal to 4 times

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5, which is equal to 20.

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Square root of 25 is 5.

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Let's do another one.

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What's 3 times the square root of 29?

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Well 29 only has two factors.

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It's a prime number.

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It only has the factors 1 and 29.

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And neither of those numbers are perfect squares.

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So we really can't simplify this one anymore.

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So this is already in completely simplified form.

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Let's do a couple more.

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What about 7 times the square root of 320?

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Let's think about 320.

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Well we could actually do it in steps when we have

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larger numbers like this.

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I can look at it and say, well it does look like 4 -- actually

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it looks like 16 would go into this because 16 goes into 32.

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So let's try that.

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So that equals 7 times the square root of 16 times 20.

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That just equals 7 times the square root of 16 times

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the square root of 20.

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7 times the square root of 16.

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The square root of 16 is 4.

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So 7 times 4 is 28.

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So that's 28 times the square root of 20.

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Now are we done?

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Well actually, I think I can factor 20 even more because

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20 is equal to 4 times 5.

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So I can say this is equal to 28 times the square

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root of 4 times 5.

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The square root of 4 is 2 so that could just take the 2 out

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and that becomes 56 times the square root of 5.

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I hope that made sense to you.

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And this is actually a pretty important technique

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I just did here.

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Immediately when I look at 320.

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I don't know what the largest number is that goes into 320.

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It actually turns out that it's 64.

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But just looking at the number, I said, well I

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know that 4 goes into it.

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So I could have just pulled out 4 and then said, "Oh, that's

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equal to 4 times 80." And then I would have had

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to work with 80.

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In this case, I saw 32 and I was like, it looks like 16 goes

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into it and I factored out 16 first and when I took out the

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square root of 16, I multiplied the outside by 4 and

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that's how I got the 28.

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But then I reduced the number on the inside said, "Oh, well

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that still is divisible by a perfect square.

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It's still divisible by 4." And then I kept doing it until I

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was left with essentially, a prime number or a number that

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couldn't be reduced anymore under the radical.

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And it actually doesn't have to be prime.

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So hopefully, that gives you a good sense of how to do

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radical simplification.

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It's really just an extension of the exponent rules that

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you've already learned and hopefully as you do the module,

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you'll get good at it.

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Have fun.

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