1
00:00:00,000 --> 00:00:07,646
We have to simplify the principal square root of -52 and we're going to assume because we have a negative

2
00:00:07,646 --> 00:00:15,210
52 here inside of the radical then this is the principal branch of the complex square root function

3
00:00:15,210 --> 00:00:19,726
that we can actually input negative numbers in the domain of this function

4
00:00:19,726 --> 00:00:24,136
that we can actually get the imaginary or complex result.

5
00:00:24,136 --> 00:00:28,344
So we can rewrite negative 52 as negative 1 times 52.

6
00:00:28,344 --> 00:00:38,134
So this can be rewritten as the principal square root of negative 1 times 52.

7
00:00:38,134 --> 00:00:43,311
And then if we assume that this is the principal branch of the complex square root function,

8
00:00:43,311 --> 00:00:50,127
we can rewrite this -- this is going to be equal to the square root of negative 1 times the principal --

9
00:00:50,127 --> 00:00:58,134
or I should've say the principal square root of negative 1 times the principal square root of 52.

10
00:00:58,134 --> 00:01:00,954
Now, I want to be very, very clear here.

11
00:01:00,954 --> 00:01:06,230
You can do what we just did if we have the principal square root of the product of two things.

12
00:01:06,230 --> 00:01:12,775
We can rewrite that as the principal square root of each and then we take the product, but you can only do this --

13
00:01:12,775 --> 00:01:19,450
or I should've say you can only do this if either both of these numbers are positive or only one of them is negative.

14
00:01:19,450 --> 00:01:23,038
You cannot do this if both of these were negative.

15
00:01:23,038 --> 00:01:25,657
For example: you cannot do this.

16
00:01:25,657 --> 00:01:37,259
You cannot say the principal square root of 52 is equal to negative 1 times negative 52.

17
00:01:37,259 --> 00:01:43,304
You could do this so far; I haven't said anything wrong. 52 is definitely negative 1 times negative 52,

18
00:01:43,304 --> 00:01:49,977
but then since these are negative, you cannot then say that this is equal to

19
00:01:49,977 --> 00:01:54,867
the square root of negative 1 times the square root of negative 52.

20
00:01:54,867 --> 00:01:57,497
In fact, I invite you to continue on this train of reasoning here,

21
00:01:57,497 --> 00:01:59,451
and we're going to get a nonsensical answer.

22
00:01:59,451 --> 00:02:06,959
This is not okay. You can not do this right over here, and the reason why you can not do this is that

23
00:02:06,959 --> 00:02:12,098
this property does not work when both of these numbers are negative.

24
00:02:12,098 --> 00:02:18,142
Now that said we can do it if only one of them are negative or both of them are positive on this thing.

25
00:02:18,142 --> 00:02:25,559
Now, the principal square root of negative 1, if we're talking about the principal branch of the complex square root function, is i.

26
00:02:25,559 --> 00:02:33,644
So this right over here does simplify to i. And let's see if we can simplify the square root of 52, any.

27
00:02:33,644 --> 00:02:36,235
And to do that, we can, well, think about its prime factorization;

28
00:02:36,235 --> 00:02:48,104
see if we have any perfect square sitting in there. So 52 is 2 times 26, and 26 is 2 times 13.

29
00:02:48,104 --> 00:02:52,455
So we have 2 times 2 there, or 4 there, which is a perfect square.

30
00:02:52,455 --> 00:02:57,702
So we can rewrite this as equal to (this is equal to) -- well, we have our i now.

31
00:02:57,702 --> 00:03:03,448
Square root of -- the principal square root of negative 1 is i; the other square root of negative 1 is negative i.

32
00:03:03,448 --> 00:03:13,438
The principal square root of negative 1 is i, and then we're going to multiply that times the square root of 4 times 13.

33
00:03:13,438 --> 00:03:26,114
4 times 13. And this is equal to i times the square root of 4 -- principal square root of 4

34
00:03:26,114 --> 00:03:33,569
times the principal square root of 13. The principal square root of 4 is 2, so this all simplifies --

35
00:03:33,569 --> 00:03:39,377
and we can switch the order over here -- this is equal to 2 times the square root of 13 --

36
00:03:39,377 --> 00:03:45,503
2 times the principal square root of 13, I should say, times i.

37
00:03:45,503 --> 00:03:51,610
And I just switched around the order; makes it a little bit more easier to read if I put i after the numbers over here,

38
00:03:51,610 --> 00:03:54,441
but I'm just multiplying i times 2 times the square root of 13;

39
00:03:54,441 --> 00:03:59,524
this is the same thing as multiplying 2 times the principal square root of 13 times i.

40
00:03:59,524 --> 00:04:03,524
And I think this is about as simplified as we can get here.