0:00:00.500,0:00:09.060
Let's see if we can simplify 5[br]times the square root of 117.

0:00:09.060,0:00:13.060
So 117 doesn't jump out at me as[br]some type of a perfect square.

0:00:13.060,0:00:14.990
So let's actually take[br]its prime factorization

0:00:14.990,0:00:20.130
and see if any of those prime[br]factors show up more than once.

0:00:20.130,0:00:21.750
So clearly it's an odd number.

0:00:21.750,0:00:24.140
It's clearly not divisible by 2.

0:00:24.140,0:00:25.727
To test whether[br]it's divisible by 3,

0:00:25.727,0:00:27.060
we can add up all of the digits.

0:00:27.060,0:00:29.810
And we explain why this works in[br]another place on Khan Academy.

0:00:29.810,0:00:31.860
But if you add up all[br]the digits, you get a 9.

0:00:31.860,0:00:36.225
And 9 is divisible by 3, so 117[br]is going to be divisible by 3.

0:00:36.225,0:00:37.600
Now, let's do a[br]little aside here

0:00:37.600,0:00:41.340
to figure out what 117[br]divided by 3 actually is.

0:00:41.340,0:00:43.700
So 3 doesn't go into 1.

0:00:43.700,0:00:46.010
It does go into 11, three times.

0:00:46.010,0:00:47.670
3 times 3 is 9.

0:00:47.670,0:00:50.390
Subtract, you got[br]a remainder of 2.

0:00:50.390,0:00:53.400
Bring down a 7.

0:00:53.400,0:00:55.850
3 goes into 27 nine times.

0:00:55.850,0:00:58.087
9 times 3 is 27.

0:00:58.087,0:00:59.170
Subtract, and you're done.

0:00:59.170,0:01:02.080
It goes in perfectly.

0:01:02.080,0:01:07.550
So we can factor[br]117 as 3 times 39.

0:01:07.550,0:01:10.935
Now 39, we can factor as--[br]that jumps out more at us

0:01:10.935,0:01:13.010
that that's divisible by 3.

0:01:13.010,0:01:15.820
That's equivalent to 3 times 13.

0:01:15.820,0:01:18.320
And then all of these[br]are now prime numbers.

0:01:18.320,0:01:23.580
So we could say that this[br]thing is the same as 5 times

0:01:23.580,0:01:34.585
the square root of[br]3 times 3 times 13.

0:01:37.061,0:01:39.560
And this is going to be the[br]same thing as-- and we know this

0:01:39.560,0:01:43.210
from our exponent[br]properties-- 5 times

0:01:43.210,0:01:54.880
the square root of 3 times 3[br]times the square root of 13.

0:01:54.880,0:01:56.744
Now, what's the square[br]root of 3 times 3?

0:01:56.744,0:01:58.160
Well, that's the[br]square root of 9.

0:01:58.160,0:01:59.730
That's the square[br]root of 3 squared.

0:01:59.730,0:02:02.120
Any of those-- well, that's[br]just going to give you 3.

0:02:02.120,0:02:04.590
So this is just going[br]to simplify to 3.

0:02:04.590,0:02:10.470
So this whole thing is 5 times[br]3 times the square root of 13.

0:02:10.470,0:02:14.750
So this part right over[br]here would give us 15 times

0:02:14.750,0:02:19.850
the square root of 13.

0:02:19.850,0:02:21.750
Let's do one more example here.

0:02:21.750,0:02:29.896
So let's try to simplify 3[br]times the square root of 26.

0:02:29.896,0:02:31.770
I'm actually going[br]to put 26 in yellow,

0:02:31.770,0:02:35.160
like I did in the[br]previous problem.

0:02:35.160,0:02:37.442
Well, 26 is clearly[br]an even number,

0:02:37.442,0:02:38.900
so it's going to[br]be divisible by 2.

0:02:38.900,0:02:41.917
We can rewrite it as 2 times 13.

0:02:41.917,0:02:42.750
And then we're done.

0:02:42.750,0:02:43.820
13 is a prime number.

0:02:43.820,0:02:45.860
We can't factor this any more.

0:02:45.860,0:02:48.204
And so 26 doesn't have[br]any perfect squares in it.

0:02:48.204,0:02:49.620
It's not like we[br]can factor it out

0:02:49.620,0:02:50.970
as a factor of[br]some other numbers

0:02:50.970,0:02:52.720
and some perfect squares[br]like we did here.

0:02:52.720,0:02:55.430
117 is 13 times 9.

0:02:55.430,0:02:58.740
It's the product of a[br]perfect square and 13.

0:02:58.740,0:03:01.645
26 isn't, so we've simplified[br]this about as much as we can.

0:03:01.645,0:03:08.138
We would just leave this as 3[br]times the square root of 26.