WEBVTT
1
00:00:00.000 --> 00:00:07.000
Simplifying square roots.
2
00:00:07.000 --> 00:00:15.000
Here we have a selection of square roots, and we're going to write them in simplest radical form.
3
00:00:15.000 --> 00:00:29.000
So, taking a look at the square root of 12, we can prime factor 12 into say 3 times 4.
4
00:00:29.000 --> 00:00:36.000
3 is a prime number, 4 factors into 2 times 2.
5
00:00:36.000 --> 00:00:47.000
That means that the square root of 12 can be written as the square root of 3 times 2 times 2.
6
00:00:47.000 --> 00:00:55.000
But we know what the square root of 2 times 2 is. 2 times 2 is another way of writing 4.
7
00:00:55.000 --> 00:01:04.000
And the square root of 4 is 2. So, the square root of 2 times 2 is 2.
8
00:01:04.000 --> 00:01:11.000
The square root of 3 is an irrational number, so we're going to leave it inside the square root sign.
9
00:01:11.000 --> 00:01:22.000
So, the simplest radical form of the square root of 12 is 2 times the square root of 3.
10
00:01:22.000 --> 00:01:26.000
Now let's look at problem B.
11
00:01:26.000 --> 00:01:31.000
We should pretty well know that the square root of 36 is 6.
12
00:01:31.000 --> 00:01:36.000
But if this method is any good at all, it should get us that same answer.
13
00:01:36.000 --> 00:01:48.000
So, doing it the same way. Prime factor, 6 times 6, 3 times 2, 3 times 2.
14
00:01:48.000 --> 00:01:58.000
So, the square root of 36 is also the square root of 3 times 2 times 3 times 2.
15
00:01:58.000 --> 00:02:07.000
But we know what the square root of 3 times 3 is. The square root of 3 times 3 is 3 because 3 times 3 is 9.
16
00:02:07.000 --> 00:02:15.000
We know what the square root of 2 times 2 is. The square root of 2 is 2.
17
00:02:15.000 --> 00:02:20.000
And it leaves nothing inside the square root sign for us to have any need of it.
18
00:02:20.000 --> 00:02:31.000
So, the answer is 3 times 2 are 6. And the square root of 36 is 6 just like we knew it was.
19
00:02:31.000 --> 00:02:38.000
Okay, looking at the next problem, the square root of 42.
20
00:02:38.000 --> 00:02:52.000
Well, 42 prime factors into 6 times 7. 6 can be further prime-factored into 2 times 3.
21
00:02:52.000 --> 00:03:02.000
So, the square root of 42 is equal to the square root of 2 times 3 times 7.
22
00:03:02.000 --> 00:03:09.000
Well, it's prime-factored. And there are no pairs. We don't know what the square root of 2 is.
23
00:03:09.000 --> 00:03:15.000
We don't know what the square root of 3 is. We don't know what the square root of 7 is except that they're all irrational numbers.
24
00:03:15.000 --> 00:03:30.000
There's nothing more to be done with this. The square root of 42 is the square root of 42 in simplest radical form.
25
00:03:30.000 --> 00:03:36.000
Now, let's look at three other problems that don't have a leading coefficient of 1.
26
00:03:36.000 --> 00:03:45.000
Leading coefficient is a 3. Well, we start out by doing it exactly the same way.
27
00:03:45.000 --> 00:03:54.000
We prime factor 18, which is 2 times 9. 9 is 3 times 3.
28
00:03:54.000 --> 00:04:11.000
So, 3 times the square root of 18 is the same as 3 times the square root of 2 times 3 times 3.
29
00:04:11.000 --> 00:04:16.000
But we know the square root of 3 times 3 is 3.
30
00:04:16.000 --> 00:04:31.000
So, that's going to be a 3 times. There was a 3 already out front times the square root of 2, which doesn't pair off nicely.
31
00:04:31.000 --> 00:04:46.000
So, the final answer is 3 times 3. 9 times the square root of 2.
32
00:04:46.000 --> 00:04:54.000
I hear two more. I want you to pause the video and give those two a try on your own.
33
00:04:54.000 --> 00:05:03.000
And then we'll come back and check them.
34
00:05:03.000 --> 00:05:07.000
Alright, let's try these.
35
00:05:07.000 --> 00:05:13.000
We know that this is going to be 5 times the square root of something.
36
00:05:13.000 --> 00:05:17.000
We have to figure out what the prime factorization of 48 is first.
37
00:05:17.000 --> 00:05:26.000
So, this is 6 times 8. 6 is 2 times 3. 8 is 2 times 4, which is 2 times 2.
38
00:05:26.000 --> 00:05:28.000
Sorry for the lack of space here.
39
00:05:28.000 --> 00:05:32.000
So, the prime factorization of 5 times the square root of 48.
40
00:05:32.000 --> 00:05:43.000
I'm sorry, the next step is 5 times the square root of 2 times 3 times 2 times 2 times 2 times 2.
41
00:05:43.000 --> 00:05:57.000
Now, we look for pairs. Our answer is going to be 5 times, let's see.
42
00:05:57.000 --> 00:06:04.000
There's a pair of twos. There's another pair of twos.
43
00:06:04.000 --> 00:06:10.000
And that's going to leave us the lone square root of 3 inside.
44
00:06:10.000 --> 00:06:22.000
5 times 2 is 10 times 2 is 20. 20 square root 3.
45
00:06:22.000 --> 00:06:27.000
Now, let's try and do the next.
46
00:06:27.000 --> 00:06:31.000
I should have left a little more space here. I'm sorry.
47
00:06:31.000 --> 00:06:39.000
So, this is going to be 6 times the square root of whatever the prime factorization of the square root of.
48
00:06:39.000 --> 00:06:46.000
Whatever the prime factorization of 150 is. Now, we should be good enough at this to do a little bit of prime factorization in our head.
49
00:06:46.000 --> 00:06:51.000
150 is 15 times 10.
50
00:06:51.000 --> 00:07:02.000
15 is 3 times 5. 10 is 2 times 5.
51
00:07:02.000 --> 00:07:07.000
15 times 10 is 150.
52
00:07:07.000 --> 00:07:14.000
Now, let's see if we have any pairs. Our answer is going to involve 6 times 5.
53
00:07:14.000 --> 00:07:21.000
Well, let's see. We have a pair of 5's. 6 times 5.
54
00:07:21.000 --> 00:07:31.000
No more nice pairs. We don't know what the square root of 3 is. We don't know what the square root of 2 is other than irrational.
55
00:07:31.000 --> 00:07:42.000
And we can further simplify that a bit. 6 times 5 is 30 times the square root of 2 times 3.
56
00:07:42.000 --> 00:08:11.000
6, 30 times the square root of 6. Hope you did well.