Simplifying square roots

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

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

So let's actually take
its prime factorization

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

So clearly it's an odd number.

It's clearly not divisible by 2.

To test whether
it's divisible by 3,

we can add up all of the digits.

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

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

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

Now, let's do a
little aside here

to figure out what 117
divided by 3 actually is.

So 3 doesn't go into 1.

It does go into 11, three times.

3 times 3 is 9.

Subtract, you got
a remainder of 2.

Bring down a 7.

3 goes into 27 nine times.

9 times 3 is 27.

Subtract, and you're done.

It goes in perfectly.

So we can factor
117 as 3 times 39.

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

that that's divisible by 3.

That's equivalent to 3 times 13.

And then all of these
are now prime numbers.

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

the square root of
3 times 3 times 13.

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

from our exponent
properties-- 5 times

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

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

Well, that's the
square root of 9.

That's the square
root of 3 squared.

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

So this is just going
to simplify to 3.

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

So this part right over
here would give us 15 times

the square root of 13.

Let's do one more example here.

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

I'm actually going
to put 26 in yellow,

like I did in the
previous problem.

Well, 26 is clearly
an even number,

so it's going to
be divisible by 2.

We can rewrite it as 2 times 13.

And then we're done.

13 is a prime number.

We can't factor this any more.

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

It's not like we
can factor it out

as a factor of
some other numbers

and some perfect squares
like we did here.

117 is 13 times 9.

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

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

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