0:00:00.000,0:00:00.630


0:00:00.630,0:00:03.160
We're asked to multiply[br]and to simplify.

0:00:03.160,0:00:05.790
And we have x squared minus[br]the principal square root

0:00:05.790,0:00:11.420
of 6 times x squared plus the[br]principal square root of 2.

0:00:11.420,0:00:13.430
And so we really just[br]have two binomials,

0:00:13.430,0:00:15.630
two two-term expressions[br]that we want to multiply,

0:00:15.630,0:00:17.290
and there's multiple[br]ways to do this.

0:00:17.290,0:00:19.100
I'll show you the[br]more intuitive way,

0:00:19.100,0:00:19.990
and then I'll show[br]you the way it's

0:00:19.990,0:00:21.420
taught in some[br]algebra classes, which

0:00:21.420,0:00:23.070
might be a little bit[br]faster, but requires

0:00:23.070,0:00:24.278
a little bit of memorization.

0:00:24.278,0:00:26.320
So I'll show you the[br]intuitive way first.

0:00:26.320,0:00:28.520
So if you have[br]anything-- so let's say

0:00:28.520,0:00:31.400
I have a times x[br]plus y-- we know

0:00:31.400,0:00:33.960
from the distributive[br]property that this

0:00:33.960,0:00:39.230
is the same thing as ax plus ay.

0:00:39.230,0:00:40.980
And so we can do the[br]same thing over here.

0:00:40.980,0:00:44.810
If you view a as x squared--[br]as this whole expression

0:00:44.810,0:00:48.400
over here-- x squared minus[br]the principal square root of 6,

0:00:48.400,0:00:51.219
and you view x plus y[br]as this thing over here,

0:00:51.219,0:00:52.010
you can distribute.

0:00:52.010,0:00:55.650


0:00:55.650,0:00:58.580
We can distribute all[br]of this onto-- let

0:00:58.580,0:01:02.880
me do it this way-- distribute[br]this entire term onto this term

0:01:02.880,0:01:04.580
and onto that term.

0:01:04.580,0:01:05.670
So let's do that.

0:01:05.670,0:01:09.470
So we get x squared minus[br]the principal square root

0:01:09.470,0:01:14.690
of 6 times this term-- I'll do[br]it in yellow-- times x squared.

0:01:14.690,0:01:17.515
And then we have plus[br]this thing again.

0:01:17.515,0:01:18.640
We're just distributing it.

0:01:18.640,0:01:19.590
It's just like they say.

0:01:19.590,0:01:20.964
It's sometimes[br]not that intuitive

0:01:20.964,0:01:22.550
because this is[br]a big expression,

0:01:22.550,0:01:25.300
but you can treat it just like[br]you would treat a variable over

0:01:25.300,0:01:25.800
here.

0:01:25.800,0:01:28.840
You're distributing it over[br]this expression over here.

0:01:28.840,0:01:33.730
And so then we have x squared[br]minus the principal square root

0:01:33.730,0:01:37.225
of 6 times the principal[br]square root of 2.

0:01:37.225,0:01:41.050


0:01:41.050,0:01:44.290
And now we can do the[br]distributive property again,

0:01:44.290,0:01:48.760
but what we'll do is we'll[br]distribute this x squared

0:01:48.760,0:01:51.810
onto each of these terms and[br]distribute the square root of 2

0:01:51.810,0:01:54.360
onto each of these terms.

0:01:54.360,0:01:55.960
It's the exact[br]same thing as here,

0:01:55.960,0:01:58.230
it's just you could imagine[br]writing it like this.

0:01:58.230,0:02:04.770
x plus y times a is still[br]going to be ax plus ay.

0:02:04.770,0:02:07.598
And just to see the pattern, how[br]this is really the same thing

0:02:07.598,0:02:09.139
as this up here,[br]we're just switching

0:02:09.139,0:02:10.620
the order of the multiplication.

0:02:10.620,0:02:13.350
You can kind of view it as we're[br]distributing from the right.

0:02:13.350,0:02:16.590
And so if you do this, you[br]get x squared times x squared,

0:02:16.590,0:02:21.060
which is x to the fourth,[br]that's that times that, and then

0:02:21.060,0:02:24.355
minus x squared times the[br]principal square root of 6.

0:02:24.355,0:02:28.300


0:02:28.300,0:02:30.980
And then over here you[br]have square root of 2 times

0:02:30.980,0:02:36.570
x squared, so plus x squared[br]times the square root of 2.

0:02:36.570,0:02:39.360
And then you have square root[br]of 2 times the square root of 6.

0:02:39.360,0:02:41.440
And we have a negative[br]sign out here.

0:02:41.440,0:02:43.106
Now if you take the[br]square root of 2--

0:02:43.106,0:02:44.980
let me do this on the[br]side-- square root of 2

0:02:44.980,0:02:48.680
times the square root of 6, we[br]know from simplifying radicals

0:02:48.680,0:02:51.780
that this is the exact same[br]thing as the square root of 2

0:02:51.780,0:02:55.502
times 6, or the principal[br]square root of 12.

0:02:55.502,0:02:57.460
So the square root of 2[br]times square root of 6,

0:02:57.460,0:02:58.835
we have a negative[br]sign out here,

0:02:58.835,0:03:02.290
it becomes minus the[br]square root of 12.

0:03:02.290,0:03:05.171
And let's see if we can[br]simplify this at all.

0:03:05.171,0:03:05.670
Let's see.

0:03:05.670,0:03:08.320
You have an x to[br]the fourth term.

0:03:08.320,0:03:10.731
And then here you[br]have-- well depending

0:03:10.731,0:03:12.730
on how you want to view[br]it, you could say, look,

0:03:12.730,0:03:14.160
we have to second degree terms.

0:03:14.160,0:03:15.860
We have something[br]times x squared,

0:03:15.860,0:03:17.970
and we have something[br]else times x squared.

0:03:17.970,0:03:19.970
So if you want,[br]you could simplify

0:03:19.970,0:03:21.820
these two terms over here.

0:03:21.820,0:03:25.436
So I have square[br]root of 2 x squareds

0:03:25.436,0:03:27.810
and then I'm going to subtract[br]from that square root of 6

0:03:27.810,0:03:28.960
x squareds.

0:03:28.960,0:03:32.490
So you could view this[br]as square root of 2

0:03:32.490,0:03:35.510
minus the square root of 6, or[br]the principal square root of 2

0:03:35.510,0:03:40.260
minus the principal square[br]root of 6, x squared.

0:03:40.260,0:03:44.010
And then, if you want,[br]square root of 12,

0:03:44.010,0:03:45.560
you might be able[br]to simplify that.

0:03:45.560,0:03:48.500
12 is the same[br]thing as 3 times 4.

0:03:48.500,0:03:52.050
So the square root of 12[br]is equal to square root

0:03:52.050,0:03:54.712
of 3 times square root of 4.

0:03:54.712,0:03:57.170
And the square root of 4, or[br]the principal square root of 4

0:03:57.170,0:03:58.761
I should say, is 2.

0:03:58.761,0:04:00.510
So the square root of[br]12 is the same thing

0:04:00.510,0:04:02.620
as 2 square roots of 3.

0:04:02.620,0:04:04.900
So instead of writing the[br]principal square root of 12,

0:04:04.900,0:04:08.900
we could write minus 2 times[br]the principal square root of 3.

0:04:08.900,0:04:13.910
And then out here you have[br]an x to the fourth plus this.

0:04:13.910,0:04:15.950
And you see, if you[br]distributed this out,

0:04:15.950,0:04:18.240
if you distribute this x[br]squared, you get this term,

0:04:18.240,0:04:19.997
negative x squared,[br]square root of 6,

0:04:19.997,0:04:22.330
and if you distribute it onto[br]this, you'd get that term.

0:04:22.330,0:04:27.310
So you could debate which[br]of these two is more simple.

0:04:27.310,0:04:29.080
Now I mentioned[br]that this way I just

0:04:29.080,0:04:30.580
did the distributive[br]property twice.

0:04:30.580,0:04:31.820
Nothing new, nothing fancy.

0:04:31.820,0:04:35.380
But in some classes, you will[br]see something called FOIL.

0:04:35.380,0:04:38.040
And I think we've done[br]this in previous videos.

0:04:38.040,0:04:39.540
FOIL.

0:04:39.540,0:04:41.500
I'm not a big fan of[br]it because it's really

0:04:41.500,0:04:44.002
a way to memorize a process[br]as opposed to understanding

0:04:44.002,0:04:46.460
that this is really just from[br]the common-sense distributive

0:04:46.460,0:04:47.269
property.

0:04:47.269,0:04:48.810
But all this is is[br]a way to make sure

0:04:48.810,0:04:50.476
that you're multiplying[br]everything times

0:04:50.476,0:04:52.570
everything when[br]you're multiplying

0:04:52.570,0:04:55.290
two binomials times[br]each other like this.

0:04:55.290,0:05:00.370
And FOIL just says, look,[br]first multiply the first term.

0:05:00.370,0:05:04.160
So x squared times x[br]squared is x to the fourth.

0:05:04.160,0:05:06.670
Then multiply the outside.

0:05:06.670,0:05:09.100
So then multiply--[br]I'll do this in green--

0:05:09.100,0:05:10.360
then multiply the outside.

0:05:10.360,0:05:14.302
So the outside terms are x[br]squared and square root of 2.

0:05:14.302,0:05:16.010
And so x squared times[br]square root of 2--

0:05:16.010,0:05:20.210
and they are positive-- so[br]plus square root of 2 times x

0:05:20.210,0:05:21.040
squared.

0:05:21.040,0:05:23.702
And then multiply the inside.

0:05:23.702,0:05:25.160
And you can see[br]why I don't like it

0:05:25.160,0:05:26.730
that much is because you[br]really don't know you're doing.

0:05:26.730,0:05:28.710
You're just applying[br]an algorithm.

0:05:28.710,0:05:30.809
Then you'll multiply the inside.

0:05:30.809,0:05:32.850
And so negative square[br]root of 6 times x squared.

0:05:32.850,0:05:36.020


0:05:36.020,0:05:40.304
And then you multiply[br]the last terms.

0:05:40.304,0:05:42.470
So negative square root of[br]6 times square root of 2,

0:05:42.470,0:05:44.490
that is-- and we[br]already know that-- that

0:05:44.490,0:05:49.020
is negative square root of[br]12, which you can also then

0:05:49.020,0:05:51.920
simplify to that expression[br]right over there.

0:05:51.920,0:05:55.910
So it's fine to use[br]this, although it's good,

0:05:55.910,0:05:58.930
even if you do use this, to[br]know where FOIL comes from.

0:05:58.930,0:06:01.640
It really just comes from[br]using the distributive property

0:06:01.640,0:06:03.190
twice.