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Welcome to the presentation
on simplifying radicals.

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So let's get started with getting a little terminology out of the way.

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You're probably just wondering what a radical is and I'll just let you know.

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I've got to get the pen settings right.

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A radical is just that.

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Or you're probably more familiar calling that the square root symbol.

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So with the terminology out of
the way,

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let's actually talk about what it means to simplify a radical.

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

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is actually making it more complicated.

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But let's see.

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Let me erase that.

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

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you'd say hey, that's easy.

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That's just equal to 6 times 6

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or you'd say the square 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, right?

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So let's write that.

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Square root of 72 is the same thing as the square root of 36 times 2.

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Right? We just rewrote seventy-two as thirty-six times two.

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And the square root, if you remember from level 3 exponents.

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square root is the same thing as something to the one half 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 radical simplification works,

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and that it's really not a new concept.

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So this is the same thing as
36 times 2 to the one half power.

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Right? Because it's just a square root
is the same thing as one half power.

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And we learned from the exponent rules that when you multiply two numbers

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and then you raise that to the one half power,

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that that's the same thing as raising each of the numbers to the one half power

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and then multiplying. Right?

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

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the square root symbol, into the one half power.

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And I did that just to show you that this is just an extension of the exponent rules.

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It isn't really a new concept.

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Although, I guess sometimes 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 become more obvious.

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The square root of 50.

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Well, the square root of 50 --

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50 is the same thing as 25 times 2.

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

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The square root of 25 times 2 is the same thing as the square root of 25

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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,

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but how did you know to split 50
into 25 and 2?"

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Why didn't I say that 50 is equal to the square root of 5 and 10?

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

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I don't know what
other factors 50 has.

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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 of 50--

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I actually wanted the largest factor of 50 that 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 done with it,

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because neither 5 nor 10 are perfect squares

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and the same thing's with 1 and 50.

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So the way you should think
about it,

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think about the factors of the original number

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and figure out if any of those 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 recognize them more readily.

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

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then you'll probably want to factor them out.

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And then you can take them out of the radical sign

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like 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,

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that just means times the square root of 27.

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

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and 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 --

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that's equal to 7 times the square root of 9 times 3.

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And now, based on the rules we just learned,

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that's the same thing as 7 times the square
root of 9

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times the square root of 3.

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

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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 nine times the square root of eighteen?

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Well, once again, what are the factors of eighteen?

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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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So 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 --

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that's a 2, 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

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27 times 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, twenty-five itself is a perfect square.

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This is kind of a problem that's so easy that 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 5,

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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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So, let's think about 320.

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Well we could actually do it in steps when we have larger numbers like this.

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

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actually 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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Well, that just equals 7 times the square root of 16

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times 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

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because 20 is equal to 4 times 5.

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So I can say this is equal to 28 times the square 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 know that 4 goes into it.

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So I could have just pulled out 4,

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and then said, "Oh, that's equal to 4 times 80."

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And then I would have had to work with 80.

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

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and I factored out 16 first.

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And when I took out the square root of 16, I multiplied the outside by 4

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and that's how I got the 28.

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But then I reduced the number
on the inside and said,

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"Oh, well that still is divisible by a perfect square.

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

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until I was left with essentially, a prime number or a number that 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 radical simplification.

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It's really just an extension of the exponent rules that you've already learned,

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and hopefully as you do the module, you'll get good at it.

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Have fun!