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We're asked to subtract all of this
craziness

2
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over here, and it looks daunting, but if

3
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we really just focus, it actually should
be

4
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pretty straightforward to subtract and
simplify this thing.

5
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Cuz right from the get go, I have 4 times
the fourth root of 81x to the fifth,

6
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and from that I wanna subtract 2 times the
fourth root of 81x to the fifth.

7
00:00:23,180 --> 00:00:26,600
And so, you really can just say, lookI
have four of something and this

8
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something, I'll just circle in yellow I
have four of this, it could be lemons.

9
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I have four of these things and I wanna
subtract two of these things.

10
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These are the exact same things.

11
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They're the 4th root of 81x to the 5th,
4th root of 81x to the 5th.

12
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So if I have four of, if I have four
lemons and

13
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I wanna subtract two lemons, I'm gonna
have two lemons left over.

14
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Or if I have four of this thing and I take
away

15
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two of this thing, I'm gonna have two of
these things left over.

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So these terms right over here simplify to
2

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times the fouthth root of 81x to the
fifth.

18
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And I got this 2 just by subtracting the
coefficients 4

19
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something minus 2 something is equal to 2
of that something.

20
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And then that, of course, we still have
this minus the regular

21
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principal square root of x to the third,
of x to the third.

22
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Now I wanted to try to simplify, I wanna
try to

23
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simplify what's inside of these under the
radical signs, so that

24
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we can, on this, in this example actually
take the fourth

25
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root and over here actually take, maybe, a
principal square root.

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So first of all, let's see if 81 either is
a, is something to the fourth power,

27
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or at least can be factored into something

28
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that is a, a something to the fourth
power.

29
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So 81, if we do prime factorization, is 3
times 27,

30
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27 is 3 times 9, and 9 is 3 times 3.

31
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So 81 is exactly 3 times 3 times 3 times
3.

32
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So 81 actually is 3 to the fourth power
which is

33
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convenient, cuz we're gonna be taking the
fourth root of that.

34
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And then x to the fifth we can write as a
product.

35
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We can, let me write it over here so it
doesn't get messy.

36
00:02:01,570 --> 00:02:05,750
So I'm gonna write what's under the
radical as 3 to

37
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the fourth power, times, times x to the
fourth power times x.

38
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x to the fourth times x is x to the fifth
power.

39
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And, I'm taking the fourth root of all of
this.

40
00:02:16,270 --> 00:02:18,330
And, taking the fourth root of all of
this,

41
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that's the same thing as taking the fourth
root

42
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of this and, taking the fourth root of
this,

43
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and let me just, I'm gonna just skip step.

44
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So, I'm taking the fourth root.

45
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I'm taking the fourth root of all of it,
right over there.

46
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And of course I have a 2 out front.

47
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And then x to the third can be written as
x squared times x.

48
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It's minus the principle square root of x
squared times x.

49
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And I broke it up like this cuz this,
right over here, is a perfect square.

50
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Now, how can we simplify this a little
bit?

51
00:02:48,570 --> 00:02:50,480
And you're probably getting used to the
pattern.

52
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This is the same thing as the fourth root
after you get the fourth,

53
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times the fourth root of x to the fourth,
times the fourth root of x.

54
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So let's just skip straight to that.

55
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So what is, what is the fourth root?

56
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Well I ca, I can write it, let me write

57
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it explicitly, although you wouldn't have
to necessarily do this.

58
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This is the same thing as the fourth, as
the fourth root of 3 to the fourth,

59
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times the fourth root of x to the fourth,
times the fourth root

60
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of x, times the fourth root of x, and 2 as
being multiplied times all of that.

61
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And then this over here is minus the
principle square

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root of x squared, times the principle
square root of x.

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And so, if we try to simplify it, the
fourth

64
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root of 3 to the fourth power is just 3.

65
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So, we get a 3 there, the fourth root of x
to the fourth power is just going to be

66
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x, is just going to be is just, actually,
look,

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I just reminded myself, you have to be
careful there.

68
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It is not just x, because what if x is
negative?

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If x is negative, then x to the fourth
power is going

70
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to be a positive value, and when you take
the fourth, remember, this

71
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is the fourth principle root, you're going
to get the positive version

72
00:04:02,830 --> 00:04:06,260
of x, or really, you're going to get the
absolute value of x.

73
00:04:06,260 --> 00:04:08,450
So here you're going to be getting, you're

74
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going to be getting the absolute value of
x.

75
00:04:12,590 --> 00:04:15,620
And then, although, well you could make an
argument

76
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that x needs to be positive if this thing

77
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is going to be well-defined in the real
numbers,

78
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cuz then what's under the radical has to
be positive.

79
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But let's just go with this for right now.

80
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And then we have the fourth root of x.

81
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And then over here the, the principal
square of

82
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x squared by the same logic, by the same
logic

83
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is going to be the absolute value of x and

84
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then this is just the principal square
root of x.

85
00:04:38,710 --> 00:04:40,030
So, let's multiply everything out.

86
00:04:40,030 --> 00:04:44,480
We have 2 times 3 times the absolute value
of x.

87
00:04:44,480 --> 00:04:50,640
So, 2 times 3 is 6 times the absolute
value of x, times the principal

88
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or the, the principal fourth root of x, I
should say, minus,

89
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minus, we've took out the absolute value
of x, times the principle root of x.

90
00:05:02,470 --> 00:05:04,390
And we can't do anymore subtracting.

91
00:05:04,390 --> 00:05:06,750
Just because you have to realize, this is
a fourth

92
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root, this is a regular square root,
principle square root.

93
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If these were the same root, then maybe we
could simplify this a little bit more.

94
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And so then we are all done, and we have
fully simplified it.

95
00:05:18,200 --> 00:05:21,130
And if you make the assumption that this
is defined

96
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for real numbers so that the domain over
here this would

97
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has to be under these radicals has to be
positive

98
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actually everyone of these cases and if
there need to be

99
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positive not gonna be dealing with
imaginary numbers all of

100
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these need to be positive, their domains
are x has to

101
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be greater than or equal to 0 then you
could assume

102
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that the absolute value of x is the same
as x.

103
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But I will just take it right here, if you
restrict

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the domain you could get rid of the
absolute value sign.