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Let's see if we can simplify 5
times the square root of 117.

3
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So 117 doesn't jump out at me as
some type of a perfect square.

4
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So let's actually take
its prime factorization

5
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and see if any of those prime
factors show up more than once.

6
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So clearly it's an odd number.

7
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It's clearly not divisible by 2.

8
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To test whether
it's divisible by 3,

9
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we can add up all of the digits.

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And we explain why this works in
another place on Khan Academy.

11
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But if you add up all
the digits, you get a 9.

12
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And 9 is divisible by 3, so 117
is going to be divisible by 3.

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Now, let's do a
little aside here

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to figure out what 117
divided by 3 actually is.

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So 3 doesn't go into 1.

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It does go into 11, three times.

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3 times 3 is 9.

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Subtract, you got
a remainder of 2.

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Bring down a 7.

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3 goes into 27 nine times.

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9 times 3 is 27.

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Subtract, and you're done.

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It goes in perfectly.

24
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So we can factor
117 as 3 times 39.

25
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Now 39, we can factor as--
that jumps out more at us

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that that's divisible by 3.

27
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That's equivalent to 3 times 13.

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And then all of these
are now prime numbers.

29
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So we could say that this
thing is the same as 5 times

30
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the square root of
3 times 3 times 13.

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And this is going to be the
same thing as-- and we know this

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from our exponent
properties-- 5 times

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the square root of 3 times 3
times the square root of 13.

35
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Now, what's the square
root of 3 times 3?

36
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Well, that's the
square root of 9.

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That's the square
root of 3 squared.

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Any of those-- well, that's
just going to give you 3.

39
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So this is just going
to simplify to 3.

40
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So this whole thing is 5 times
3 times the square root of 13.

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So this part right over
here would give us 15 times

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the square root of 13.

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Let's do one more example here.

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So let's try to simplify 3
times the square root of 26.

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I'm actually going
to put 26 in yellow,

46
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like I did in the
previous problem.

47
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Well, 26 is clearly
an even number,

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so it's going to
be divisible by 2.

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We can rewrite it as 2 times 13.

50
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And then we're done.

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13 is a prime number.

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We can't factor this any more.

53
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And so 26 doesn't have
any perfect squares in it.

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It's not like we
can factor it out

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as a factor of
some other numbers

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and some perfect squares
like we did here.

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117 is 13 times 9.

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It's the product of a
perfect square and 13.

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26 isn't, so we've simplified
this about as much as we can.

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We would just leave this as 3
times the square root of 26.

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