1
00:00:00,000 --> 00:00:00,440


2
00:00:00,440 --> 00:00:05,900
We're asked to simplify d times 4d to the 1/2 power.

3
00:00:05,900 --> 00:00:07,000
So let's see what we can do here.

4
00:00:07,000 --> 00:00:09,539
And I think there's multiple ways to kind of simplify it,

5
00:00:09,539 --> 00:00:11,259
and we'll explore all of them.

6
00:00:11,259 --> 00:00:13,150
So d times 4d to the 1/2.

7
00:00:13,150 --> 00:00:17,060
4d to the 1/2, let me just focus on that right there.

8
00:00:17,059 --> 00:00:18,489
4d to the 1/2.

9
00:00:18,489 --> 00:00:21,359
If you take the product of two things and you raise that to

10
00:00:21,359 --> 00:00:25,109
an exponent, that's the same thing as taking each of those

11
00:00:25,109 --> 00:00:27,949
terms to that exponent and then taking the product.

12
00:00:27,949 --> 00:00:33,210
So 4d to the 1/2 is the same thing as 4 to the 1/2 times d

13
00:00:33,210 --> 00:00:34,259
to the 1/2.

14
00:00:34,259 --> 00:00:39,189
And of course, you have this d out front.

15
00:00:39,189 --> 00:00:40,669
And I could put the parentheses here, but it

16
00:00:40,670 --> 00:00:41,730
doesn't matter at this point.

17
00:00:41,729 --> 00:00:43,250
Now you just have three things that you're

18
00:00:43,250 --> 00:00:44,590
taking the product of.

19
00:00:44,590 --> 00:00:47,310
So I'll put this d out front just like that.

20
00:00:47,310 --> 00:00:49,230
And now there's a couple of ways you could think about it.

21
00:00:49,229 --> 00:00:52,129
You have this 4 to the 1/2 power and you're taking its

22
00:00:52,130 --> 00:00:54,650
principal square root, so its positive square root.

23
00:00:54,649 --> 00:00:57,030
So that right there is going to be 2.

24
00:00:57,030 --> 00:00:59,780
So 4 to the 1/2, that right there is going

25
00:00:59,780 --> 00:01:02,789
to be equal to 2.

26
00:01:02,789 --> 00:01:05,179
And then, there's a couple of ways you could think about it.

27
00:01:05,180 --> 00:01:08,740
You have this d out front.

28
00:01:08,739 --> 00:01:09,989
And then you have a d to the 1/2.

29
00:01:09,989 --> 00:01:12,659


30
00:01:12,659 --> 00:01:15,439
And there's several ways that you can now represent this,

31
00:01:15,439 --> 00:01:18,640
depending on what you feel-- I don't know--

32
00:01:18,640 --> 00:01:20,120
aesthetically pleasant.

33
00:01:20,120 --> 00:01:23,109
You could say, well look, this is d to the first power.

34
00:01:23,109 --> 00:01:24,590
This is d to the 1/2 power.

35
00:01:24,590 --> 00:01:27,579
I could add them.

36
00:01:27,579 --> 00:01:32,329
So one way to say it is it's 2 times d to the 1 plus 1/2.

37
00:01:32,329 --> 00:01:34,629
You can either say 1 and 1/2 power or 3/2

38
00:01:34,629 --> 00:01:36,109
power if you want.

39
00:01:36,109 --> 00:01:38,810
So that's one simplification of this expression.

40
00:01:38,810 --> 00:01:41,469
Another simplification would be well, you could say, look.

41
00:01:41,469 --> 00:01:46,280
d to the 1/2, that is the same thing as the square root of d.

42
00:01:46,280 --> 00:01:51,650
So you could say this is the same thing as 2 times d times

43
00:01:51,650 --> 00:01:53,670
the square root of d.

44
00:01:53,670 --> 00:01:56,570
So any of these are actual simplifications, but it's

45
00:01:56,569 --> 00:01:58,029
interesting to see that they're really all

46
00:01:58,030 --> 00:02:00,799
representing the same thing.

47
00:02:00,799 --> 00:02:01,200