0:00:00.610,0:00:03.300
We're asked to subtract all of this[br]craziness

0:00:03.300,0:00:05.630
over here, and it looks daunting, but if

0:00:05.630,0:00:08.460
we really just focus, it actually should[br]be

0:00:08.460,0:00:11.055
pretty straightforward to subtract and[br]simplify this thing.

0:00:11.055,0:00:17.080
Cuz right from the get go, I have 4 times[br]the fourth root of 81x to the fifth,

0:00:17.080,0:00:23.180
and from that I wanna subtract 2 times the[br]fourth root of 81x to the fifth.

0:00:23.180,0:00:26.600
And so, you really can just say, lookI[br]have four of something and this

0:00:26.600,0:00:30.460
something, I'll just circle in yellow I[br]have four of this, it could be lemons.

0:00:30.460,0:00:33.800
I have four of these things and I wanna[br]subtract two of these things.

0:00:33.800,0:00:35.840
These are the exact same things.

0:00:35.840,0:00:39.650
They're the 4th root of 81x to the 5th,[br]4th root of 81x to the 5th.

0:00:39.650,0:00:42.530
So if I have four of, if I have four[br]lemons and

0:00:42.530,0:00:46.170
I wanna subtract two lemons, I'm gonna[br]have two lemons left over.

0:00:46.170,0:00:47.810
Or if I have four of this thing and I take[br]away

0:00:47.810,0:00:51.740
two of this thing, I'm gonna have two of[br]these things left over.

0:00:51.740,0:00:56.032
So these terms right over here simplify to[br]2

0:00:56.032,0:01:00.450
times the fouthth root of 81x to the[br]fifth.

0:01:00.450,0:01:03.310
And I got this 2 just by subtracting the[br]coefficients 4

0:01:03.310,0:01:07.300
something minus 2 something is equal to 2[br]of that something.

0:01:07.300,0:01:11.560
And then that, of course, we still have[br]this minus the regular

0:01:11.560,0:01:16.160
principal square root of x to the third,[br]of x to the third.

0:01:16.160,0:01:18.450
Now I wanted to try to simplify, I wanna[br]try to

0:01:18.450,0:01:21.660
simplify what's inside of these under the[br]radical signs, so that

0:01:21.660,0:01:24.190
we can, on this, in this example actually[br]take the fourth

0:01:24.190,0:01:29.240
root and over here actually take, maybe, a[br]principal square root.

0:01:29.240,0:01:33.610
So first of all, let's see if 81 either is[br]a, is something to the fourth power,

0:01:33.610,0:01:35.210
or at least can be factored into something

0:01:35.210,0:01:37.770
that is a, a something to the fourth[br]power.

0:01:37.770,0:01:42.610
So 81, if we do prime factorization, is 3[br]times 27,

0:01:42.610,0:01:45.510
27 is 3 times 9, and 9 is 3 times 3.

0:01:45.510,0:01:49.610
So 81 is exactly 3 times 3 times 3 times[br]3.

0:01:49.610,0:01:53.010
So 81 actually is 3 to the fourth power[br]which is

0:01:53.010,0:01:56.310
convenient, cuz we're gonna be taking the[br]fourth root of that.

0:01:56.310,0:01:59.580
And then x to the fifth we can write as a[br]product.

0:01:59.580,0:02:01.570
We can, let me write it over here so it[br]doesn't get messy.

0:02:01.570,0:02:05.750
So I'm gonna write what's under the[br]radical as 3 to

0:02:05.750,0:02:11.290
the fourth power, times, times x to the[br]fourth power times x.

0:02:11.290,0:02:13.710
x to the fourth times x is x to the fifth[br]power.

0:02:13.710,0:02:16.270
And, I'm taking the fourth root of all of[br]this.

0:02:16.270,0:02:18.330
And, taking the fourth root of all of[br]this,

0:02:18.330,0:02:20.360
that's the same thing as taking the fourth[br]root

0:02:20.360,0:02:23.720
of this and, taking the fourth root of[br]this,

0:02:23.720,0:02:25.330
and let me just, I'm gonna just skip step.

0:02:25.330,0:02:27.500
So, I'm taking the fourth root.

0:02:27.500,0:02:30.730
I'm taking the fourth root of all of it,[br]right over there.

0:02:30.730,0:02:32.930
And of course I have a 2 out front.

0:02:32.930,0:02:36.380
And then x to the third can be written as[br]x squared times x.

0:02:36.380,0:02:41.540
It's minus the principle square root of x[br]squared times x.

0:02:41.540,0:02:45.780
And I broke it up like this cuz this,[br]right over here, is a perfect square.

0:02:45.780,0:02:48.570
Now, how can we simplify this a little[br]bit?

0:02:48.570,0:02:50.480
And you're probably getting used to the[br]pattern.

0:02:50.480,0:02:54.220
This is the same thing as the fourth root[br]after you get the fourth,

0:02:54.220,0:02:58.410
times the fourth root of x to the fourth,[br]times the fourth root of x.

0:02:58.410,0:02:59.970
So let's just skip straight to that.

0:02:59.970,0:03:02.798
So what is, what is the fourth root?

0:03:02.798,0:03:04.490
Well I ca, I can write it, let me write

0:03:04.490,0:03:06.380
it explicitly, although you wouldn't have[br]to necessarily do this.

0:03:06.380,0:03:12.740
This is the same thing as the fourth, as[br]the fourth root of 3 to the fourth,

0:03:12.740,0:03:18.270
times the fourth root of x to the fourth,[br]times the fourth root

0:03:18.270,0:03:24.480
of x, times the fourth root of x, and 2 as[br]being multiplied times all of that.

0:03:24.480,0:03:27.550
And then this over here is minus the[br]principle square

0:03:27.550,0:03:32.300
root of x squared, times the principle[br]square root of x.

0:03:32.300,0:03:34.110
And so, if we try to simplify it, the[br]fourth

0:03:34.110,0:03:37.550
root of 3 to the fourth power is just 3.

0:03:37.550,0:03:43.830
So, we get a 3 there, the fourth root of x[br]to the fourth power is just going to be

0:03:43.830,0:03:49.610
x, is just going to be is just, actually,[br]look,

0:03:49.610,0:03:51.150
I just reminded myself, you have to be[br]careful there.

0:03:51.150,0:03:54.410
It is not just x, because what if x is[br]negative?

0:03:54.410,0:03:56.720
If x is negative, then x to the fourth[br]power is going

0:03:56.720,0:03:59.600
to be a positive value, and when you take[br]the fourth, remember, this

0:03:59.600,0:04:02.830
is the fourth principle root, you're going[br]to get the positive version

0:04:02.830,0:04:06.260
of x, or really, you're going to get the[br]absolute value of x.

0:04:06.260,0:04:08.450
So here you're going to be getting, you're

0:04:08.450,0:04:12.590
going to be getting the absolute value of[br]x.

0:04:12.590,0:04:15.620
And then, although, well you could make an[br]argument

0:04:15.620,0:04:18.829
that x needs to be positive if this thing

0:04:18.829,0:04:21.250
is going to be well-defined in the real[br]numbers,

0:04:21.250,0:04:23.130
cuz then what's under the radical has to[br]be positive.

0:04:23.130,0:04:25.770
But let's just go with this for right now.

0:04:25.770,0:04:28.570
And then we have the fourth root of x.

0:04:28.570,0:04:30.870
And then over here the, the principal[br]square of

0:04:30.870,0:04:34.550
x squared by the same logic, by the same[br]logic

0:04:34.550,0:04:36.440
is going to be the absolute value of x and

0:04:36.440,0:04:38.710
then this is just the principal square[br]root of x.

0:04:38.710,0:04:40.030
So, let's multiply everything out.

0:04:40.030,0:04:44.480
We have 2 times 3 times the absolute value[br]of x.

0:04:44.480,0:04:50.640
So, 2 times 3 is 6 times the absolute[br]value of x, times the principal

0:04:50.640,0:04:56.490
or the, the principal fourth root of x, I[br]should say, minus,

0:04:56.490,0:05:02.470
minus, we've took out the absolute value[br]of x, times the principle root of x.

0:05:02.470,0:05:04.390
And we can't do anymore subtracting.

0:05:04.390,0:05:06.750
Just because you have to realize, this is[br]a fourth

0:05:06.750,0:05:10.710
root, this is a regular square root,[br]principle square root.

0:05:10.710,0:05:14.870
If these were the same root, then maybe we[br]could simplify this a little bit more.

0:05:14.870,0:05:18.200
And so then we are all done, and we have[br]fully simplified it.

0:05:18.200,0:05:21.130
And if you make the assumption that this[br]is defined

0:05:21.130,0:05:24.250
for real numbers so that the domain over[br]here this would

0:05:24.250,0:05:27.170
has to be under these radicals has to be[br]positive

0:05:27.170,0:05:29.860
actually everyone of these cases and if[br]there need to be

0:05:29.860,0:05:32.350
positive not gonna be dealing with[br]imaginary numbers all of

0:05:32.350,0:05:35.050
these need to be positive, their domains[br]are x has to

0:05:35.050,0:05:36.980
be greater than or equal to 0 then you[br]could assume

0:05:36.980,0:05:39.430
that the absolute value of x is the same[br]as x.

0:05:39.430,0:05:41.390
But I will just take it right here, if you[br]restrict

0:05:41.390,0:05:43.950
the domain you could get rid of the[br]absolute value sign.