Solving radical equations 3

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.050
We're asked to solve for x.

00:00:02.050 --> 00:00:05.340
So we have the square root of
the entire quantity 5x squared

00:00:05.340 --> 00:00:08.360
minus 8 is equal to 2x.

00:00:08.360 --> 00:00:10.830
Now we already have an
expression under a radical

00:00:10.830 --> 00:00:13.860
isolated, so the easiest first
step here is really just to

00:00:13.860 --> 00:00:16.500
square both sides of
this equation.

00:00:16.500 --> 00:00:21.650
So let's just square both
sides of that equation.

00:00:21.650 --> 00:00:24.950
Now the left-hand side, if you
square it, the square root of

00:00:24.950 --> 00:00:27.060
5x squared minus 8 squared
is going to be 5x

00:00:27.060 --> 00:00:28.590
squared minus 8.

00:00:28.590 --> 00:00:31.960
This is 5x squared minus 8.

00:00:31.960 --> 00:00:35.220
And then the right-hand side,
2x squared is the same thing

00:00:35.220 --> 00:00:39.530
as 2 squared times x squared
or 4x squared.

00:00:39.530 --> 00:00:42.060
Now we have a quadratic.

00:00:42.060 --> 00:00:44.590
Now let's see what we can do
to maybe simplify this a

00:00:44.590 --> 00:00:45.230
little bit more.

00:00:45.230 --> 00:00:48.345
Well, we could subtract
4x squared.

00:00:48.345 --> 00:00:51.130
Or actually, even better, let's
subtract 5x squared from

00:00:51.130 --> 00:00:53.480
both sides so that we just have
all our x terms on the

00:00:53.480 --> 00:00:54.780
right-hand side.

00:00:54.780 --> 00:00:58.000
So let's subtract 5x squared
from both sides.

00:00:58.000 --> 00:01:03.600
Subtract 5x squared from both
sides of the equation.

00:01:03.600 --> 00:01:06.180
The left-hand side,
this cancels out.

00:01:06.180 --> 00:01:06.980
That was the whole point.

00:01:06.980 --> 00:01:11.990
We're just left with negative 8
is equal to 4x squared minus

00:01:11.990 --> 00:01:13.915
5x squared, that's negative
1x squared.

00:01:13.915 --> 00:01:16.740
Or we could just write
negative x

00:01:16.740 --> 00:01:19.040
squared, just like that.

00:01:19.040 --> 00:01:21.590
And then we could multiply both
sides of this equation by

00:01:21.590 --> 00:01:22.880
negative 1.

00:01:22.880 --> 00:01:25.080
That'll make it into
positive 8.

00:01:25.080 --> 00:01:26.800
Or I could divide by negative
1, however you

00:01:26.800 --> 00:01:27.330
want to view it.

00:01:27.330 --> 00:01:30.900
Negative 1 times that
times negative 1.

00:01:30.900 --> 00:01:35.510
So we get positive 8 is
equal to x squared.

00:01:35.510 --> 00:01:38.110
And now we could take the square
root of both sides of

00:01:38.110 --> 00:01:39.360
this equation.

00:01:39.360 --> 00:01:41.760
So let's take the square
root of both

00:01:41.760 --> 00:01:43.860
sides of this equation.

00:01:43.860 --> 00:01:47.200
The principal square root of
both sides of this equation.

00:01:47.200 --> 00:01:48.400
And what do we get?

00:01:48.400 --> 00:01:51.240
We get, on the right-hand
side, x is equal to the

00:01:51.240 --> 00:01:52.670
square root of 8.

00:01:52.670 --> 00:02:00.270
And 8 can be rewritten
as 2 times 4.

00:02:00.270 --> 00:02:04.350
And this can be rewritten as the
square root of 2 times the

00:02:04.350 --> 00:02:07.460
square root of 4
is equal to x.

00:02:07.460 --> 00:02:09.470
I don't like this green
color so much.

00:02:09.470 --> 00:02:11.530
And what's the square root of
4, the principal root of 4?

00:02:11.530 --> 00:02:12.500
It's 2.

00:02:12.500 --> 00:02:14.270
So that right there is 2.

00:02:14.270 --> 00:02:20.300
So this side becomes 2, this 2,
times the square root of 2.

00:02:20.300 --> 00:02:23.850
And that is equal to x.

00:02:23.850 --> 00:02:26.970
Now let's verify that this
is the solution to

00:02:26.970 --> 00:02:28.350
our original equation.

00:02:28.350 --> 00:02:31.260
So let's substitute this in,
first to the left-hand side.

00:02:31.260 --> 00:02:37.270
So on the left-hand side, we
have 5 times 2 square roots of

00:02:37.270 --> 00:02:42.860
2 squared minus 8.

00:02:42.860 --> 00:02:44.320
And then we're going to have
to take the square root of

00:02:44.320 --> 00:02:46.100
that whole thing.

00:02:46.100 --> 00:02:47.880
So this is going to be equal
to-- we're just focused on the

00:02:47.880 --> 00:02:49.430
left-hand side right now.

00:02:49.430 --> 00:02:55.390
This is equal to the square
root of 5 times 2 squared,

00:02:55.390 --> 00:02:59.810
which is 4, times the square
root of 2 squared, which is 2.

00:02:59.810 --> 00:03:01.190
And then minus 8.

00:03:01.190 --> 00:03:08.100
And this is 5 times 4
is 20 times 2 is 40.

00:03:08.100 --> 00:03:10.760
And then you have 40
minus 8 is 32.

00:03:10.760 --> 00:03:14.000
So this is equal to the
square root of 32.

00:03:14.000 --> 00:03:15.723
Square root of 32 is the same
thing as the square

00:03:15.723 --> 00:03:17.480
root of 16 times 2.

00:03:17.480 --> 00:03:19.205
The square root of 16 is 4.

00:03:19.205 --> 00:03:21.680
So this is the same thing as
the square root of 16 times

00:03:21.680 --> 00:03:23.000
the square root of 2.

00:03:23.000 --> 00:03:25.670
Or 4 square roots of 2.

00:03:25.670 --> 00:03:29.750
So that's what the left hand
simplifies to when we-- and

00:03:29.750 --> 00:03:31.400
remember, the original equation
didn't have these

00:03:31.400 --> 00:03:33.770
squares here, so if you just
look at the green part, the

00:03:33.770 --> 00:03:36.590
green part on the left-hand
side just simplified to 4

00:03:36.590 --> 00:03:37.650
roots of 2.

00:03:37.650 --> 00:03:39.700
Let's see what 2x
simplifies to.

00:03:39.700 --> 00:03:42.570
Our original right-hand
side was just the 2x.

00:03:42.570 --> 00:03:45.360
That's parentheses with the
square added later.

00:03:45.360 --> 00:03:46.790
So what's 2x?

00:03:46.790 --> 00:03:49.620
2 times 2 roots of 2.

00:03:49.620 --> 00:03:51.270
2 square root of 2.

00:03:51.270 --> 00:03:54.660
Well that's just 4
square root of 2.

00:03:54.660 --> 00:03:57.370
So when x is equal to 2 square
roots of 2, the left-hand side

00:03:57.370 --> 00:03:58.425
equals 4 square roots of 2.

00:03:58.425 --> 00:04:00.380
And remember, the left-hand side
looked like this when we

00:04:00.380 --> 00:04:01.500
started off.

00:04:01.500 --> 00:04:03.140
The left-hand side when
we started off

00:04:03.140 --> 00:04:04.930
didn't have that there.

00:04:04.930 --> 00:04:07.220
I want to make that clear.

00:04:07.220 --> 00:04:09.980
So when you substitute this back
into this left-hand side,

00:04:09.980 --> 00:04:11.370
you get 4 square roots of 2.

00:04:11.370 --> 00:04:13.510
When you substitute it back into
the original right-hand

00:04:13.510 --> 00:04:14.970
side, you get 4 square
roots of 2.

00:04:14.970 --> 00:04:18.450
So it's definitely the right--
I'm trying to write in black.

00:04:18.450 --> 00:04:21.149
It's definitely the
right solution.


Solving radical equations 2

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:02.450
We're asked to solve for y.

00:00:02.450 --> 00:00:05.750
So we're told that the negative
of the cube root of y

00:00:05.750 --> 00:00:09.770
is equal to 4 times the
cube root of y plus 5.

00:00:09.770 --> 00:00:11.790
So in all of these it's helpful
to just be able to

00:00:11.790 --> 00:00:15.400
isolate the cube root, isolate
the radical in the equation,

00:00:15.400 --> 00:00:16.270
and then solve from there.

00:00:16.270 --> 00:00:18.960
So let's see if we can
isolate the radical.

00:00:18.960 --> 00:00:22.360
So the simplest thing to do, if
we want all of the radical

00:00:22.360 --> 00:00:25.750
onto the left-hand side
equation, we can subtract 4

00:00:25.750 --> 00:00:28.355
times the cube root of y from
both sides of this equation.

00:00:28.355 --> 00:00:33.570
So let's subtract 4.

00:00:33.570 --> 00:00:38.280
We want to subtract 4 times the
cube root of y from both

00:00:38.280 --> 00:00:39.530
sides of this equation.

00:00:43.750 --> 00:00:49.110
And so your left-hand side, you
already have negative 1

00:00:49.110 --> 00:00:51.600
times the cube root of y, and
you're going to subtract 4

00:00:51.600 --> 00:00:53.480
more of the cube root of y.

00:00:53.480 --> 00:00:56.610
So you're going to have
negative 5 times the

00:00:56.610 --> 00:00:59.130
cube root of y.

00:00:59.130 --> 00:01:00.690
That's your left-hand side.

00:01:00.690 --> 00:01:03.420
Now the right-hand side-- these
two guys-- cancel out.

00:01:03.420 --> 00:01:06.220
That was the whole point behind
subtracting this value.

00:01:06.220 --> 00:01:09.880
So that cancels out and you're
just left with a 5 there.

00:01:09.880 --> 00:01:13.460
You're just left with this
5 right over there.

00:01:13.460 --> 00:01:16.855
Now, we've almost isolated
this cube root of y.

00:01:16.855 --> 00:01:18.900
We just have to divide both
sides of the equation by

00:01:18.900 --> 00:01:20.600
negative 5.

00:01:20.600 --> 00:01:22.220
So you just divide both
sides of this

00:01:22.220 --> 00:01:25.330
equation by negative 5.

00:01:25.330 --> 00:01:26.620
And these cancel out.

00:01:26.620 --> 00:01:27.700
That was the whole point.

00:01:27.700 --> 00:01:33.460
And we are left with the cube
root of y is equal to-- 5

00:01:33.460 --> 00:01:36.920
divided by negative
5 is negative 1.

00:01:36.920 --> 00:01:40.570
Now, the cube root of y is
equal to negative 1.

00:01:40.570 --> 00:01:43.440
Well the easiest way to solve
this is, let's take both sides

00:01:43.440 --> 00:01:46.420
of this equation to
the third power.

00:01:46.420 --> 00:01:50.810
This statement right here is
the exact same statement as

00:01:50.810 --> 00:01:54.840
saying y to the 1/3 is
equal to negative 1.

00:01:54.840 --> 00:01:56.330
These are just two
different ways of

00:01:56.330 --> 00:01:57.240
writing the same thing.

00:01:57.240 --> 00:01:59.970
This is equivalent to taking
the 1/3 power.

00:01:59.970 --> 00:02:02.150
So if we take both sides of
this equation to the third

00:02:02.150 --> 00:02:09.160
power, that's like taking both
sides of this equation to the

00:02:09.160 --> 00:02:10.410
third power.

00:02:12.890 --> 00:02:16.390
And you can see here, y to the
1/3 to the third-- y to the

00:02:16.390 --> 00:02:18.760
1/3 and then to the third,
that's like saying y to the

00:02:18.760 --> 00:02:20.830
1/3 times 3 power.

00:02:20.830 --> 00:02:22.320
Or y to the first power.

00:02:22.320 --> 00:02:23.520
That's the whole point of it.

00:02:23.520 --> 00:02:25.840
If you take the cube root of y
to the third power, that's

00:02:25.840 --> 00:02:27.580
just going to be y.

00:02:27.580 --> 00:02:29.610
So the left-hand
side becomes y.

00:02:29.610 --> 00:02:31.190
And then the right-hand side,
what's negative 1

00:02:31.190 --> 00:02:31.850
to the third power?

00:02:31.850 --> 00:02:33.700
Negative 1 times negative
1 is 1.

00:02:33.700 --> 00:02:35.950
Times negative 1 again
is negative 1.

00:02:35.950 --> 00:02:40.790
So we get y is equal to negative
1 as our solution.

00:02:40.790 --> 00:02:42.460
Now let's make sure that
it actually works.

00:02:42.460 --> 00:02:45.395
Let's go back to our
original equation.

00:02:45.395 --> 00:02:49.170
And I'll put negative
1 in for our y's.

00:02:49.170 --> 00:02:53.550
We had the negative of the
cube root of-- this time,

00:02:53.550 --> 00:02:59.510
negative 1-- has to be equal
to 4 times the cube root of

00:02:59.510 --> 00:03:01.560
negative 1 plus 5.

00:03:01.560 --> 00:03:03.750
Let's verify that this
is the case.

00:03:03.750 --> 00:03:07.210
The cube root of negative
1 is negative 1.

00:03:07.210 --> 00:03:08.770
Negative 1 to the third
power is negative 1.

00:03:08.770 --> 00:03:13.230
So this is equal to the negative
of negative 1 has to

00:03:13.230 --> 00:03:17.110
be equal to 4 times-- the cube
root of negative 1 is

00:03:17.110 --> 00:03:19.870
negative 1 plus 5.

00:03:19.870 --> 00:03:22.330
The negative of negative
1 is just positive 1.

00:03:22.330 --> 00:03:25.120
So 1 needs to be equal to--
4 times negative 1,

00:03:25.120 --> 00:03:27.070
negative 4, plus 5.

00:03:27.070 --> 00:03:28.110
This is true.

00:03:28.110 --> 00:03:30.260
Negative 4 plus 5 is 1.

00:03:30.260 --> 00:03:31.420
So this works out.

00:03:31.420 --> 00:03:33.490
This is our solution.


Simplifying Radical Expressions 2

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:05.790
Simplify 3 times the square
root of 200x to the third.

00:00:05.790 --> 00:00:09.130
So a good place to start,
let's factor out 200.

00:00:09.130 --> 00:00:15.170
So if I'm just focusing on the
200 here, 200 is 2 times 100.

00:00:15.170 --> 00:00:16.390
And 100 is a perfect square.

00:00:16.390 --> 00:00:19.950
It's 10 times 10.

00:00:19.950 --> 00:00:21.910
And then if we were to factor
x to the third, that's the

00:00:21.910 --> 00:00:24.695
same thing as x squared
times x.

00:00:24.695 --> 00:00:26.060
So let's write that out.

00:00:26.060 --> 00:00:27.080
So let me just rewrite.

00:00:27.080 --> 00:00:31.210
This whole thing is equal to 3
times the square root-- I'll

00:00:31.210 --> 00:00:35.480
write it all under one big
radical sign right now-- of 2

00:00:35.480 --> 00:00:41.890
times 100 times x
squared times x.

00:00:41.890 --> 00:00:43.610
Stretch the radical sign.

00:00:43.610 --> 00:00:47.490
So the x squared times x,
that is x to the third.

00:00:47.490 --> 00:00:52.070
And then the 200
is 2 times 100.

00:00:52.070 --> 00:00:55.370
Now, we can you think of this
as all of these numbers, the

00:00:55.370 --> 00:00:57.860
product of all these numbers to
the 1/2 power or taking the

00:00:57.860 --> 00:01:00.320
square root of the product,
that's like taking the square

00:01:00.320 --> 00:01:03.130
root of-- or that's equivalent
to-- taking the square root of

00:01:03.130 --> 00:01:09.090
each of these terms. So this
is going to be equal to 3

00:01:09.090 --> 00:01:16.240
times the square root of 2 times
the square root of 100

00:01:16.240 --> 00:01:21.370
times the square root of
x squared times the

00:01:21.370 --> 00:01:22.620
square root of x.

00:01:25.650 --> 00:01:27.200
These terms right there
in blue, that's

00:01:27.200 --> 00:01:29.000
these terms right there.

00:01:29.000 --> 00:01:32.200
The terms in orange, that's
those terms right over there.

00:01:32.200 --> 00:01:35.170
Now, several of these are not
too hard to evaluate.

00:01:35.170 --> 00:01:37.370
So we have our 3 out front.

00:01:37.370 --> 00:01:39.410
So this is going to
be equal to 3.

00:01:39.410 --> 00:01:41.990
And then let me mark off the
terms. Let's do this one the

00:01:41.990 --> 00:01:42.460
first.

00:01:42.460 --> 00:01:44.030
What's the square root of 100?

00:01:44.030 --> 00:01:45.880
Or the principal root of 100?

00:01:45.880 --> 00:01:46.890
The positive square root.

00:01:46.890 --> 00:01:47.580
Well it's 10.

00:01:47.580 --> 00:01:49.160
10 times 10 is 100.

00:01:49.160 --> 00:01:50.950
It's going to be 3 times 10.

00:01:50.950 --> 00:01:52.570
That's that term right there.

00:01:52.570 --> 00:01:56.570
And then, what is the square
root of x squared?

00:01:56.570 --> 00:01:59.790
Well it's just x. x times
x is x squared.

00:01:59.790 --> 00:02:01.840
So times x.

00:02:01.840 --> 00:02:04.130
And then we have to deal-- so
we've dealt with that, we've

00:02:04.130 --> 00:02:05.390
dealt with that, we've
dealt with that.

00:02:05.390 --> 00:02:07.940
And then we have these two
terms right here where we

00:02:07.940 --> 00:02:09.080
can't take the square roots.

00:02:09.080 --> 00:02:11.320
So we could say times the square
root of 2 times the

00:02:11.320 --> 00:02:13.100
square root of x.

00:02:13.100 --> 00:02:17.180
So this term right here times
the square root of 2.

00:02:17.180 --> 00:02:22.860
And then, finally, out here
times the square root of x.

00:02:22.860 --> 00:02:28.990
So if we want to simplify this
fully, we'd get 3 times 10

00:02:28.990 --> 00:02:31.540
times x, which is 30x.

00:02:31.540 --> 00:02:34.090
And then we could actually
merge these.

00:02:34.090 --> 00:02:35.960
The square root of 2 times
the square root of x.

00:02:35.960 --> 00:02:38.950
That's the same thing as
the square root of 2x.

00:02:38.950 --> 00:02:44.890
So we could say times the
square root of 2x.

00:02:44.890 --> 00:02:48.980
And we have simplified this
as far as it can go.


Simplifying Radical Expressions 1

WEBVTT
Kind: captions
Language: en

00:00:00.940 --> 00:00:06.380
So we need to find the cube
root of negative 343.

00:00:06.380 --> 00:00:08.700
Another way to think about this,
this is the same thing

00:00:08.700 --> 00:00:15.170
as negative 343 to
the 1/3 power.

00:00:15.170 --> 00:00:20.510
Or, another way of thinking
about it, if x to the third is

00:00:20.510 --> 00:00:27.030
equal to negative 343-- so let
me write if x to the third is

00:00:27.030 --> 00:00:34.600
equal to negative 343, then x
is equal to the cube root of

00:00:34.600 --> 00:00:36.740
negative 343.

00:00:36.740 --> 00:00:39.120
So these are all ways
of viewing it.

00:00:39.120 --> 00:00:41.530
This is essentially saying,
there's some number, when I

00:00:41.530 --> 00:00:44.510
multiply it by itself three
times, I should

00:00:44.510 --> 00:00:46.100
get negative 343.

00:00:46.100 --> 00:00:48.390
So let's figure out what
this number is.

00:00:48.390 --> 00:00:50.890
And the easiest way I can think
of doing it is to do a

00:00:50.890 --> 00:00:53.660
prime factorization of negative
343, including

00:00:53.660 --> 00:00:56.900
factoring out the negative
numbers or a negative 1.

00:00:56.900 --> 00:00:58.960
So if we factor out the negative
1, the first thing we

00:00:58.960 --> 00:01:03.350
get, it's negative
1 times 343.

00:01:03.350 --> 00:01:06.430
Let's think what other prime
numbers go into 343.

00:01:06.430 --> 00:01:07.430
2 doesn't go into it.

00:01:07.430 --> 00:01:08.840
This is an odd number.

00:01:08.840 --> 00:01:10.230
3 doesn't go into it.

00:01:10.230 --> 00:01:13.200
If you add these numbers,
it adds up to 7.

00:01:13.200 --> 00:01:14.590
That's not divisible by 3.

00:01:14.590 --> 00:01:18.930
That's a quick test for testing
divisibility by 3.

00:01:18.930 --> 00:01:19.920
5 doesn't go into it.

00:01:19.920 --> 00:01:22.640
It doesn't end with
a 5 or a 0 here.

00:01:22.640 --> 00:01:24.520
7 is the next prime number.

00:01:24.520 --> 00:01:27.690
Let's see if it goes into 343.

00:01:27.690 --> 00:01:35.280
7 goes into 343-- 7 goes
into 34 four times.

00:01:35.280 --> 00:01:38.390
4 times 7 is 28.

00:01:38.390 --> 00:01:39.480
Subtract.

00:01:39.480 --> 00:01:41.780
34 minus 28 is 6.

00:01:41.780 --> 00:01:43.640
Bring down the 3.

00:01:43.640 --> 00:01:47.090
7 goes into 63 nine times.

00:01:47.090 --> 00:01:49.330
9 times 7 is 63.

00:01:49.330 --> 00:01:50.060
Subtract.

00:01:50.060 --> 00:01:50.780
We have no remainder.

00:01:50.780 --> 00:01:53.710
So it goes in exactly
49 times.

00:01:53.710 --> 00:01:58.720
So 343 is the same thing
as 7 times 49.

00:01:58.720 --> 00:02:04.030
And 49 is 7 times 7.

00:02:04.030 --> 00:02:06.790
So we can rewrite this
expression up here.

00:02:06.790 --> 00:02:15.010
We can rewrite it as the cube
root of negative 1 times 7

00:02:15.010 --> 00:02:17.430
times 7 times 7.

00:02:17.430 --> 00:02:20.020
Or another way to think about
it, we could view this as

00:02:20.020 --> 00:02:25.680
being equal to the cube root of
negative 1 times the cube

00:02:25.680 --> 00:02:30.960
root of 7 times 7 times 7.

00:02:30.960 --> 00:02:32.770
Now what's the cube root
of negative 1?

00:02:32.770 --> 00:02:35.790
What number to the third power
is equal to negative 1?

00:02:35.790 --> 00:02:38.960
Well, negative 1 to the third
power is negative 1, right?

00:02:38.960 --> 00:02:45.280
Negative 1 times negative 1
times negative 1 is equal to

00:02:45.280 --> 00:02:46.500
negative 1, right?

00:02:46.500 --> 00:02:47.330
This is equal to 1.

00:02:47.330 --> 00:02:49.590
Then times negative 1 is equal
to negative 1 again.

00:02:49.590 --> 00:02:51.070
So that's negative 1.

00:02:51.070 --> 00:02:53.730
And then what's the
second part?

00:02:53.730 --> 00:02:57.550
What times itself three times is
equal to 7 times 7 times 7?

00:02:57.550 --> 00:02:58.880
Well 7.

00:02:58.880 --> 00:03:01.140
This is literally 7 times
itself three times.

00:03:01.140 --> 00:03:02.930
So the second part is 7.

00:03:02.930 --> 00:03:06.880
So the answer's going to be
negative 1 times 7, or

00:03:06.880 --> 00:03:09.580
negative 7.

00:03:09.580 --> 00:03:10.830
And you can verify
for yourself.

00:03:10.830 --> 00:03:13.630
You take negative 7, multiply
it by itself 3 times.

00:03:16.300 --> 00:03:18.040
The negative part, you could
view as just negative 1 to the

00:03:18.040 --> 00:03:18.500
third power.

00:03:18.500 --> 00:03:19.910
We'll end up with a negative.

00:03:19.910 --> 00:03:23.490
And then the 7 to the third
power will give you the 343.


Pythagorean theorem 3

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:04.640
A carpet measures 7 feet long
and has a diagonal measurement

00:00:04.640 --> 00:00:07.530
of square root of 74 feet.

00:00:07.530 --> 00:00:09.845
Find the width of the carpet.

00:00:09.845 --> 00:00:12.630
So let's draw ourselves
a carpet here.

00:00:12.630 --> 00:00:15.080
So let's draw a carpet.

00:00:15.080 --> 00:00:18.410
It has a length of 7 feet, so
let's say that that is 7 feet,

00:00:18.410 --> 00:00:19.350
right there.

00:00:19.350 --> 00:00:21.620
And it's going to be a rectangle
of some kind.

00:00:21.620 --> 00:00:23.450
So let's say that we're
looking down on

00:00:23.450 --> 00:00:25.350
the carpet like that.

00:00:25.350 --> 00:00:27.340
That's our carpet.

00:00:27.340 --> 00:00:29.280
And then it has a diagonal
measurement of

00:00:29.280 --> 00:00:32.590
square root of 74 feet.

00:00:32.590 --> 00:00:35.570
So that means that this
distance, right here-- draw it

00:00:35.570 --> 00:00:39.280
a little bit neater than that
--this distance right here,

00:00:39.280 --> 00:00:45.000
the diagonal of the carpet, is
the square root of 74 feet.

00:00:45.000 --> 00:00:47.850
And what they want to know is
the width of the carpet.

00:00:47.850 --> 00:00:49.580
Find the width of the carpet.

00:00:49.580 --> 00:00:52.640
So let's say that this is
the width of the carpet.

00:00:52.640 --> 00:00:55.220
That is w, right there.

00:00:55.220 --> 00:00:59.300
Now, you might already realize
that what I have drawn here is

00:00:59.300 --> 00:01:00.520
a right triangle.

00:01:00.520 --> 00:01:02.310
Let me make sure
you realize it.

00:01:02.310 --> 00:01:04.910
This is a 90 degree
angle here.

00:01:04.910 --> 00:01:08.660
And since that is a triangle
that has a 90 degree angle,

00:01:08.660 --> 00:01:10.050
it's a right triangle.

00:01:10.050 --> 00:01:14.520
The side opposite the right
angle, or the 90 degrees, is a

00:01:14.520 --> 00:01:16.370
hypotenuse, or the
longest side.

00:01:16.370 --> 00:01:18.520
It is the square root of 74.

00:01:18.520 --> 00:01:21.710
And the shorter sides
are w and 7.

00:01:21.710 --> 00:01:24.830
And the Pythagorean Theorem
tells us that the sum of the

00:01:24.830 --> 00:01:27.980
squares of the shorter side will
be equal to the square of

00:01:27.980 --> 00:01:30.930
the hypotenuse, so the square
of the longer side.

00:01:30.930 --> 00:01:34.930
So we get w squared,
this side squared.

00:01:34.930 --> 00:01:41.640
plus 7 squared, this other side
squared, is going to be

00:01:41.640 --> 00:01:43.790
equal to the hypotenuse
squared,

00:01:43.790 --> 00:01:47.890
square root of 74 squared.

00:01:47.890 --> 00:01:54.912
And then we get w squared plus
49 is equal to the square root

00:01:54.912 --> 00:01:56.600
of 74 squared.

00:01:56.600 --> 00:01:58.415
Well, that's just
going to be 74.

00:01:58.415 --> 00:02:00.880
It is equal to 74.

00:02:00.880 --> 00:02:03.930
We can subtract 49 from both
sides of this equation.

00:02:03.930 --> 00:02:06.790
So we have just a w squared
on the left-hand side.

00:02:06.790 --> 00:02:09.940
Subtract 49 from both sides.

00:02:09.940 --> 00:02:12.470
The left side-- these guys are
going to cancel out, we're

00:02:12.470 --> 00:02:16.800
just going to be left with a w
squared --is equal to-- What's

00:02:16.800 --> 00:02:19.900
74 minus 49?

00:02:19.900 --> 00:02:23.270
74 minus 49, well, we can do a
little bit of regrouping or

00:02:23.270 --> 00:02:25.140
borrowing here, if we don't
want to do it in our head.

00:02:25.140 --> 00:02:27.250
We can make this a 14.

00:02:27.250 --> 00:02:28.810
This becomes a 6.

00:02:28.810 --> 00:02:31.640
14 minus 9 is 5.

00:02:31.640 --> 00:02:33.850
6 minus 4 is 2.

00:02:33.850 --> 00:02:36.070
And we have w squared
is equal to 25.

00:02:36.070 --> 00:02:39.810
So w is going to be equal to
the square root of 25, the

00:02:39.810 --> 00:02:41.260
positive square root.

00:02:41.260 --> 00:02:44.200
So let's take the square root
of both sides, the positive

00:02:44.200 --> 00:02:49.870
square root, and we will
get w is equal to 5.

00:02:49.870 --> 00:02:52.210
Because we obviously we don't
want it to be negative 5.

00:02:52.210 --> 00:02:54.130
That wouldn't be a realistic
distance.

00:02:54.130 --> 00:02:57.330
So the width of the
carpet is 5.

00:02:57.330 --> 00:02:58.820
And we're done.


Pythagorean theorem 2

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:03.330
Say we have a right triangle.

00:00:03.330 --> 00:00:07.890
Let me draw my right triangle
just like that.

00:00:07.890 --> 00:00:09.530
This is a right triangle.

00:00:09.530 --> 00:00:11.650
This is the 90 degree
angle right here.

00:00:11.650 --> 00:00:14.100
And we're told that
this side's length

00:00:14.100 --> 00:00:16.440
right here is 14.

00:00:16.440 --> 00:00:19.700
This side's length right
over here is 9.

00:00:19.700 --> 00:00:22.890
And we're told that
this side is a.

00:00:22.890 --> 00:00:26.030
And we need to find
the length of a.

00:00:26.030 --> 00:00:28.860
So as I mentioned already,
this is a right triangle.

00:00:28.860 --> 00:00:33.180
And we know that if we have a
right triangle, if we know two

00:00:33.180 --> 00:00:36.110
of the sides, we can always
figure out a third side using

00:00:36.110 --> 00:00:38.290
the Pythagorean theorem.

00:00:38.290 --> 00:00:41.000
And what the Pythagorean theorem
tells us is that the

00:00:41.000 --> 00:00:44.400
sum of the squares of the
shorter sides is going to be

00:00:44.400 --> 00:00:47.260
equal to the square of the
longer side, or the square of

00:00:47.260 --> 00:00:48.360
the hypotenuse.

00:00:48.360 --> 00:00:49.820
And if you're not sure about
that, you're probably

00:00:49.820 --> 00:00:52.690
thinking, hey Sal, how do I know
that a is shorter than

00:00:52.690 --> 00:00:53.890
this side over here?

00:00:53.890 --> 00:00:56.220
How do I know it's
not 15 or 16?

00:00:56.220 --> 00:00:58.920
And the way to tell is that the
longest side in a right

00:00:58.920 --> 00:01:01.190
triangle, and this only applies
to a right triangle,

00:01:01.190 --> 00:01:04.180
is the side opposite the
90 degree angle.

00:01:04.180 --> 00:01:07.450
And in this case, 14 is opposite
the 90 degrees.

00:01:07.450 --> 00:01:11.930
This 90 degree angle kind of
opens into this longest side.

00:01:11.930 --> 00:01:14.170
The side that we call
the hypotenuse.

00:01:14.170 --> 00:01:16.440
So now that we know that that's
the longest side, let

00:01:16.440 --> 00:01:17.150
me color code it.

00:01:17.150 --> 00:01:19.010
So this is the longest side.

00:01:19.010 --> 00:01:21.180
This is one of the
shorter sides.

00:01:21.180 --> 00:01:23.900
And this is the other of
the shorter sides.

00:01:23.900 --> 00:01:27.910
The Pythagorean theorem tells us
that the sum of the squares

00:01:27.910 --> 00:01:35.940
of the shorter sides, so a
squared plus 9 squared is

00:01:35.940 --> 00:01:39.420
going to be equal
to 14 squared.

00:01:39.420 --> 00:01:41.870
And it's really important that
you realize that it's not 9

00:01:41.870 --> 00:01:44.070
squared plus 14 squared is
going to be equal to a

00:01:44.070 --> 00:01:46.480
squared. a squared is one
of the shorter sides.

00:01:46.480 --> 00:01:48.920
The sum of the squares of these
two sides are going to

00:01:48.920 --> 00:01:52.950
be equal to 14 squared, the
hypotenuse squared.

00:01:52.950 --> 00:01:56.060
And from here, we just
have to solve for a.

00:01:56.060 --> 00:02:05.440
So we get a squared plus 81
is equal to 14 squared.

00:02:05.440 --> 00:02:07.200
In case we don't know
what that is, let's

00:02:07.200 --> 00:02:08.060
just multiply it out.

00:02:08.060 --> 00:02:10.419
14 times 14.

00:02:10.419 --> 00:02:12.910
4 times 4 is 16.

00:02:12.910 --> 00:02:17.590
4 times 1 is 4 plus 1 is 5.

00:02:17.590 --> 00:02:18.430
Take a 0 there.

00:02:18.430 --> 00:02:20.270
1 times 4 is 4.

00:02:20.270 --> 00:02:22.390
1 times 1 is 1.

00:02:22.390 --> 00:02:24.070
6 plus 0 is 6.

00:02:24.070 --> 00:02:26.840
5 plus 4 is 9, bring
down the 1.

00:02:26.840 --> 00:02:28.490
It's 196.

00:02:28.490 --> 00:02:34.180
So a squared plus 81 is equal
to 14 squared, which is 196.

00:02:34.180 --> 00:02:36.780
Then we could subtract 81 from
both sides of this equation.

00:02:40.990 --> 00:02:43.650
On the left-hand side, we're
going to be left

00:02:43.650 --> 00:02:44.640
with just the a squared.

00:02:44.640 --> 00:02:48.120
These two guys cancel out, the
whole point of subtracting 81.

00:02:48.120 --> 00:02:56.230
So we're left with a squared
is equal to 196 minus 81.

00:02:56.230 --> 00:02:57.480
What is that?

00:02:59.590 --> 00:03:01.370
If you just subtract
1, it's 195.

00:03:01.370 --> 00:03:05.880
If you subtract 80,
it would be 115 if

00:03:05.880 --> 00:03:07.130
I'm doing that right.

00:03:09.510 --> 00:03:12.420
And then to solve for a, we just
take the square root of

00:03:12.420 --> 00:03:14.860
both sides, the principal square
root, the positive

00:03:14.860 --> 00:03:17.140
square root of both sides
of this equation.

00:03:17.140 --> 00:03:18.360
So let's do that.

00:03:18.360 --> 00:03:19.960
Because we're dealing with
distances, you can't have a

00:03:19.960 --> 00:03:22.840
negative square root, or a
negative distance here.

00:03:22.840 --> 00:03:31.140
And we get a is equal to
the square root of 115.

00:03:31.140 --> 00:03:34.170
Let's see if we can break
down 115 any further.

00:03:34.170 --> 00:03:34.980
So let's see.

00:03:34.980 --> 00:03:38.080
It's clearly divisible by 5.

00:03:38.080 --> 00:03:40.900
If you factor it out, it's
5, and then 5 goes

00:03:40.900 --> 00:03:45.270
in the 115 23 times.

00:03:45.270 --> 00:03:46.920
So both of these are
prime numbers.

00:03:46.920 --> 00:03:47.410
So we're done.

00:03:47.410 --> 00:03:49.670
So you actually can't
factor this anymore.

00:03:49.670 --> 00:03:54.800
So a is just going to be equal
to the square root of 115.

00:03:54.800 --> 00:03:57.330
Now if you want to get a sense
of roughly how large the

00:03:57.330 --> 00:04:00.280
square root of 115 is, if you
think about it, the square

00:04:00.280 --> 00:04:03.370
root of 100 is equal to 10.

00:04:03.370 --> 00:04:08.020
And the square root of
121 is equal to 11.

00:04:08.020 --> 00:04:11.050
So this value right here is
going to be someplace in

00:04:11.050 --> 00:04:14.180
between 10 and 11, which
makes sense if you

00:04:14.180 --> 00:04:16.140
think about it visually.


Pythagorean theorem 1

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.410
The main mast of a fishing boat
is supported by a sturdy

00:00:04.410 --> 00:00:07.720
rope that extends from the top
of the mast to the deck.

00:00:07.720 --> 00:00:10.990
If the mast is 20 feet tall and
the rope attaches to the

00:00:10.990 --> 00:00:14.390
deck 15 feet away from
the base of the mast,

00:00:14.390 --> 00:00:16.110
how long is the rope?

00:00:16.110 --> 00:00:19.260
So let's draw ourselves a boat
and make sure we understand

00:00:19.260 --> 00:00:22.150
what the deck and the mast
and all of that is.

00:00:22.150 --> 00:00:25.620
So let me draw a boat.

00:00:25.620 --> 00:00:27.200
I'll start with yellow.

00:00:27.200 --> 00:00:31.270
So let's say that
this is my boat.

00:00:31.270 --> 00:00:33.740
That is the deck of the boat.

00:00:33.740 --> 00:00:37.100
And the boat might look
something like this.

00:00:37.100 --> 00:00:40.480
It's a sailing boat.

00:00:40.480 --> 00:00:42.210
This is the water down here.

00:00:45.600 --> 00:00:47.875
And then the mast is the thing
that holds up the sail.

00:00:47.875 --> 00:00:56.630
So let me draw ourselves
a mast.

00:00:56.630 --> 00:00:59.550
And they say the mast
is 20 feet tall.

00:00:59.550 --> 00:01:04.769
So this distance right
here is 20 feet.

00:01:04.769 --> 00:01:07.100
That is what is holding
up the sail.

00:01:07.100 --> 00:01:10.230
I can draw it as a pole so it's
a little bit clearer.

00:01:10.230 --> 00:01:12.960
Even shade it in if we like.

00:01:12.960 --> 00:01:16.200
And then they say a rope
attaches to the deck 15 feet

00:01:16.200 --> 00:01:18.670
away from the base of the mast.
So this is the base of

00:01:18.670 --> 00:01:20.960
the mast. This is the
deck right here.

00:01:20.960 --> 00:01:26.190
The rope attaches 15 feet away
from the base of the mast.

00:01:26.190 --> 00:01:29.590
So if this is the base of the
mast, we go 15 feet, might be

00:01:29.590 --> 00:01:31.255
about that distance
right there.

00:01:31.255 --> 00:01:32.505
Let me mark that.

00:01:36.330 --> 00:01:37.620
And the rope attaches
right here.

00:01:37.620 --> 00:01:40.340
From the top of the mast
all the way that base.

00:01:40.340 --> 00:01:44.290
So the rope goes like that.

00:01:44.290 --> 00:01:48.330
And then they ask us, how
long is the rope?

00:01:48.330 --> 00:01:50.470
So there's a few things
you might realize.

00:01:50.470 --> 00:01:52.550
We're dealing with
a triangle here.

00:01:52.550 --> 00:01:53.750
And it's not any triangle.

00:01:53.750 --> 00:01:57.680
We're assuming that the mast
goes straight up and that the

00:01:57.680 --> 00:01:59.370
deck is straight
left and right.

00:01:59.370 --> 00:02:00.970
So this is a right triangle.

00:02:00.970 --> 00:02:03.090
This is a 90 degree
angle right here.

00:02:05.870 --> 00:02:08.660
And we know that, if we know two
sides of a right triangle,

00:02:08.660 --> 00:02:11.600
we can always figure out the
third side of a right triangle

00:02:11.600 --> 00:02:12.850
using the Pythagorean theorem.

00:02:19.750 --> 00:02:24.220
And all that tells us is it the
sum of the squares of the

00:02:24.220 --> 00:02:27.610
shorter sides of the triangle
are going to be equal to the

00:02:27.610 --> 00:02:29.280
square of the longer side.

00:02:29.280 --> 00:02:30.540
And that longer side is
call the hypotenuse.

00:02:36.750 --> 00:02:39.010
And in all cases, the hypotenuse
is the side

00:02:39.010 --> 00:02:40.360
opposite the 90 degree angle.

00:02:40.360 --> 00:02:42.720
It is always going to
be the longest side

00:02:42.720 --> 00:02:44.760
of our right triangle.

00:02:44.760 --> 00:02:46.580
So we need to figure out
the hypotenuse here.

00:02:46.580 --> 00:02:48.920
We know the lengths of the
two shorter sides.

00:02:48.920 --> 00:02:54.210
So we can see that if we take 15
squared, that's one of the

00:02:54.210 --> 00:02:56.280
short sides, I'm squaring it.

00:02:56.280 --> 00:02:59.050
And then add that to the square
of the other shorter

00:02:59.050 --> 00:03:01.125
side, to 20 feet squared.

00:03:01.125 --> 00:03:03.620
And when I say the shorter side,
I mean relative to the

00:03:03.620 --> 00:03:04.260
hypotenuse.

00:03:04.260 --> 00:03:06.700
The hypotenuse will always
be the longest side.

00:03:09.660 --> 00:03:12.670
Let's say the hypotenuse is in
green just so we get our color

00:03:12.670 --> 00:03:13.940
coding nice.

00:03:13.940 --> 00:03:17.260
That is going to be equal
to the rope squared.

00:03:17.260 --> 00:03:18.520
Or the length of the rope.

00:03:18.520 --> 00:03:21.070
Let's call this distance
right here r.

00:03:21.070 --> 00:03:22.320
r for rope.

00:03:28.520 --> 00:03:30.440
So 15 squared plus 20
squared is going to

00:03:30.440 --> 00:03:33.080
be equal to r squared.

00:03:33.080 --> 00:03:34.510
And what's 15 squared?

00:03:34.510 --> 00:03:37.260
It's 225.

00:03:37.260 --> 00:03:43.840
20 squared is 400.

00:03:43.840 --> 00:03:47.460
And that's going to be
equal to r squared.

00:03:47.460 --> 00:03:52.260
Now 225 plus 400 is 625.

00:03:52.260 --> 00:03:55.430
625 is equal to r squared.

00:03:55.430 --> 00:03:57.443
And then we can take the
principal root of both sides

00:03:57.443 --> 00:03:58.430
of this equation.

00:03:58.430 --> 00:04:00.370
Because we're talking about
distances, we want the

00:04:00.370 --> 00:04:01.800
positive square root.

00:04:01.800 --> 00:04:04.460
So you take the positive square
root, or the principal

00:04:04.460 --> 00:04:07.140
root, of both sides
of this equation.

00:04:07.140 --> 00:04:10.050
And you are left with
r is equal to the

00:04:10.050 --> 00:04:11.540
square root of 625.

00:04:11.540 --> 00:04:13.960
You can play with it a little
bit if you like.

00:04:13.960 --> 00:04:17.500
But if you've ever played with
numbers around 25, you'll see

00:04:17.500 --> 00:04:20.740
that this is 25 squared.

00:04:20.740 --> 00:04:25.520
So r is equal to the square
root of 625, which is 25.

00:04:25.520 --> 00:04:28.160
So this distance right here,
the length of the rope, is

00:04:28.160 --> 00:04:31.700
equal to 25 feet.


Simplifying expressions with exponents 3

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:06.140
Find the volume of a cube with
a side length of 8xy squared.

00:00:06.140 --> 00:00:07.600
So we have a cube right here.

00:00:07.600 --> 00:00:10.190
And they're saying that each of
the sides, for a cube all

00:00:10.190 --> 00:00:12.430
the sides have the same length,
each of them have

00:00:12.430 --> 00:00:15.240
length 8xy squared.

00:00:15.240 --> 00:00:17.980
So the length of this side,
maybe we call this the width,

00:00:17.980 --> 00:00:21.770
is 8xy squared.

00:00:21.770 --> 00:00:24.660
The length of this side, maybe
we can call that the depth, or

00:00:24.660 --> 00:00:26.210
the length, depending
on how you want to

00:00:26.210 --> 00:00:29.460
label it, is 8xy squared.

00:00:29.460 --> 00:00:35.870
The height of this cube is also
going to be 8xy squared.

00:00:35.870 --> 00:00:38.540
A cube, all of the dimensions
are the same.

00:00:38.540 --> 00:00:41.770
Now, if you want to take the
volume of any a rectangular

00:00:41.770 --> 00:00:44.170
prism, you'll multiplied
these three dimensions.

00:00:44.170 --> 00:00:46.200
You"ll multiply the
width times the

00:00:46.200 --> 00:00:48.390
depth times the height.

00:00:48.390 --> 00:00:52.070
Now, in a cube, these are
all the same dimensions.

00:00:52.070 --> 00:00:54.460
So you can just take this
dimension and multiply it by

00:00:54.460 --> 00:00:55.940
itself three times.

00:00:55.940 --> 00:00:58.740
Or you could take it
to the third power.

00:00:58.740 --> 00:01:06.710
So the volume of this cube is
8xy squared, I could just

00:01:06.710 --> 00:01:10.760
multiply it by itself three
times, or I could say to the

00:01:10.760 --> 00:01:12.130
third power.

00:01:12.130 --> 00:01:14.120
That's why raising something
to the third power is

00:01:14.120 --> 00:01:15.510
called cubing it.

00:01:15.510 --> 00:01:18.060
Because you're essentially
finding the volume of a cube,

00:01:18.060 --> 00:01:20.550
where the thing that you're
cubing is the dimension of

00:01:20.550 --> 00:01:22.160
each of the sides.

00:01:22.160 --> 00:01:23.910
Now, what's this going to be?

00:01:23.910 --> 00:01:26.610
Well, when you take a product
and you raise it to some

00:01:26.610 --> 00:01:31.950
power, that's the same thing as
taking each of the terms to

00:01:31.950 --> 00:01:32.760
that power.

00:01:32.760 --> 00:01:39.230
So this is the same thing as 8
to the third power times x to

00:01:39.230 --> 00:01:46.120
the third power times y squared
to the third power.

00:01:49.910 --> 00:01:53.760
And what's 8 to the
third power?

00:01:53.760 --> 00:01:59.690
8 squared is 64, so we'll
multiply 64 times 8.

00:01:59.690 --> 00:02:02.670
4 times 8 is 32.

00:02:02.670 --> 00:02:05.570
6 times 8 is 48 plus 3 is 51.

00:02:05.570 --> 00:02:07.430
So it's 512.

00:02:07.430 --> 00:02:14.650
So this is equal to 512 times
x to the third and y squared

00:02:14.650 --> 00:02:17.760
to the third power, that's the
same thing is y to the 2 times

00:02:17.760 --> 00:02:20.105
3 power, or y to the
sixth power.

00:02:24.490 --> 00:02:25.050
And we're done.

00:02:25.050 --> 00:02:26.380
We found the volume
of the cube.

00:02:26.380 --> 00:02:30.710
It's 512 x to the third
y to the sixth.


Simplifying expressions with exponents 2

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.810
We're asked to simplify m to
the third and then that

00:00:03.810 --> 00:00:07.600
squared all over mx to the
third all of that to the

00:00:07.600 --> 00:00:08.580
fourth power.

00:00:08.580 --> 00:00:10.330
And they want us to
express our answer

00:00:10.330 --> 00:00:11.490
in exponential form.

00:00:11.490 --> 00:00:14.090
And then they tell us that m
does not equal zero and x does

00:00:14.090 --> 00:00:15.110
not equal zero.

00:00:15.110 --> 00:00:17.560
Because if they did, then this
denominator would be equal to

00:00:17.560 --> 00:00:19.310
zero if either of them
did and then that

00:00:19.310 --> 00:00:20.590
would just be undefined.

00:00:20.590 --> 00:00:24.600
So that's what they have to put
that little caveat there.

00:00:24.600 --> 00:00:28.170
That neither of them are going
to be equal to zero.

00:00:28.170 --> 00:00:29.890
So let's try to simplify this.

00:00:29.890 --> 00:00:30.690
Let me rewrite it.

00:00:30.690 --> 00:00:35.690
So the numerator is m to
the third squared.

00:00:35.690 --> 00:00:47.520
And our denominator is mx to the
third to the fourth power.

00:00:47.520 --> 00:00:49.490
Now let's just work on each
of these independently.

00:00:49.490 --> 00:00:53.530
What is m to the third and then
that to the second power?

00:00:53.530 --> 00:00:56.150
Well if you raise something to
an exponent, and then raise

00:00:56.150 --> 00:00:58.560
that whole thing to another
exponent, this is going to be

00:00:58.560 --> 00:00:59.730
the same thing.

00:00:59.730 --> 00:01:02.720
So this numerator up here
is going to be m to

00:01:02.720 --> 00:01:05.180
the 3 times 2 power.

00:01:05.180 --> 00:01:07.340
Or m to the sixth power.

00:01:07.340 --> 00:01:09.670
We're raising it to third
and then squaring it.

00:01:09.670 --> 00:01:12.400
And if you think about it, this
expression right here--

00:01:12.400 --> 00:01:13.950
let me just do a little
aside here.

00:01:13.950 --> 00:01:17.340
M to the third squared, there's
no magic behind why

00:01:17.340 --> 00:01:18.280
were multiplying that.

00:01:18.280 --> 00:01:20.890
If you think about m to the
third squared, this is the

00:01:20.890 --> 00:01:26.620
same thing as m to the third
times m to the third.

00:01:26.620 --> 00:01:28.050
And if you saw something
like this, you would

00:01:28.050 --> 00:01:29.160
add these two exponents.

00:01:29.160 --> 00:01:29.860
3 plus 3.

00:01:29.860 --> 00:01:32.140
You have exactly two 3's here.

00:01:32.140 --> 00:01:34.330
If this was 4, you would be
multiplying it four times.

00:01:34.330 --> 00:01:35.710
So you'd have four 3's.

00:01:35.710 --> 00:01:37.380
So that's why we're multiplying
these two

00:01:37.380 --> 00:01:39.720
numbers, 3 times 2.

00:01:39.720 --> 00:01:41.780
So that gives us
m to the sixth.

00:01:41.780 --> 00:01:45.660
And then the denominator here,
we have mx to the third to the

00:01:45.660 --> 00:01:46.920
fourth power.

00:01:46.920 --> 00:01:50.230
Now in general, when you have
the product of some numbers,

00:01:50.230 --> 00:01:54.720
and you're raising the entire
product to some exponent, you

00:01:54.720 --> 00:01:56.750
can raise each of the terms
to that exponent.

00:01:56.750 --> 00:02:00.610
So this denominator right here,
this is going to be m to

00:02:00.610 --> 00:02:05.200
the fourth times x to the
third to the fourth.

00:02:05.200 --> 00:02:08.674
I'm just raising both of these
terms, m to the fourth x to

00:02:08.674 --> 00:02:10.669
the third, to the
fourth power.

00:02:10.669 --> 00:02:12.700
Now, what does this
simplify to?

00:02:12.700 --> 00:02:16.200
Well, our numerator is
still m to the sixth.

00:02:16.200 --> 00:02:19.370
I'll do one step at a time.

00:02:19.370 --> 00:02:22.135
Our denominator is
m to the fourth.

00:02:22.135 --> 00:02:25.920
But we have x to the third and
then that to the fourth power.

00:02:25.920 --> 00:02:27.040
So we can multiply these.

00:02:27.040 --> 00:02:28.230
3 times 4.

00:02:28.230 --> 00:02:33.650
So this is x to the 12th power,
so times x to the 12th.

00:02:33.650 --> 00:02:37.130
Now, we have an m to the sixth
in the numerator and an m to

00:02:37.130 --> 00:02:38.750
the fourth in the denominator.

00:02:38.750 --> 00:02:42.400
You can view this as the same
thing as m to the sixth times

00:02:42.400 --> 00:02:44.540
m to the negative four,
either way.

00:02:44.540 --> 00:02:52.110
But we know, or hopefully we
know, that this right here can

00:02:52.110 --> 00:02:56.130
be simplified as m to the
sixth minus 4 power.

00:02:56.130 --> 00:02:58.580
So this is going to be equal
to-- so the numerator, we

00:02:58.580 --> 00:03:04.600
could have m to the sixth minus
4, which is m squared.

00:03:04.600 --> 00:03:06.480
That's what this
simplifies to.

00:03:06.480 --> 00:03:13.600
And the denominator we still
have an x to the 12th.

00:03:13.600 --> 00:03:14.460
And we're done.

00:03:14.460 --> 00:03:17.020
If we don't want something in
the denominator, we could

00:03:17.020 --> 00:03:20.500
rewrite this x to the 12th in
the denominator as x to the

00:03:20.500 --> 00:03:21.890
negative 12th in
the numerator.

00:03:21.890 --> 00:03:25.660
So we could rewrite this whole
expression as being equal to m

00:03:25.660 --> 00:03:31.390
squared times x to
the negative 12.

00:03:31.390 --> 00:03:34.180
Either of these would be
an acceptable answer.


Simplifying expressions with exponents

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:04.230
Simplify n squared r
to the fifth n to

00:00:04.230 --> 00:00:05.680
the negative seventh.

00:00:05.680 --> 00:00:09.040
Express your answer in
exponential form

00:00:09.040 --> 00:00:11.260
with positive exponents.

00:00:11.260 --> 00:00:13.120
Let me just rewrite what
they have up there.

00:00:13.120 --> 00:00:18.290
So we have n squared
r to the fifth n to

00:00:18.290 --> 00:00:19.540
the negative seventh.

00:00:19.540 --> 00:00:22.620
So the easiest way to simplify
this, I would say kind of just

00:00:22.620 --> 00:00:25.330
while we were multiplying them,
group the common bases.

00:00:25.330 --> 00:00:30.240
So we have two exponent
expressions with n as a base.

00:00:30.240 --> 00:00:33.360
We have n squared and we have
n to the negative seventh.

00:00:33.360 --> 00:00:36.230
So let me just rearrange this
multiplication right here.

00:00:36.230 --> 00:00:40.500
I can rewrite this as n squared
n to the negative

00:00:40.500 --> 00:00:44.940
seven and then we have
our r to the fifth.

00:00:44.940 --> 00:00:46.930
And what's this going
to be equal to?

00:00:46.930 --> 00:00:50.200
Well we have n squared times
n to the negative seven.

00:00:50.200 --> 00:00:52.650
We can add those
two exponents.

00:00:52.650 --> 00:00:56.840
So this right here would
simplify to n

00:00:56.840 --> 00:00:59.300
to the 2 minus 7.

00:00:59.300 --> 00:01:01.600
2 minus 7 is negative 5.

00:01:01.600 --> 00:01:05.850
N to the negative 5,
r to the fifth.

00:01:05.850 --> 00:01:09.660
Now we've simplified this
about as much as we can.

00:01:09.660 --> 00:01:11.240
But, we're not done yet.

00:01:11.240 --> 00:01:13.470
Because they said express your
answer in exponential form

00:01:13.470 --> 00:01:16.240
with positive exponents.

00:01:16.240 --> 00:01:18.350
This is a negative
exponent here.

00:01:18.350 --> 00:01:20.610
This is not acceptable,
at least based on what

00:01:20.610 --> 00:01:22.680
they asked us to do.

00:01:22.680 --> 00:01:26.760
So we know that n to the
negative fifth is the same

00:01:26.760 --> 00:01:29.510
thing as 1 over n
to the fifth.

00:01:29.510 --> 00:01:31.720
So we can just rewrite
it the that way.

00:01:31.720 --> 00:01:37.100
So this is the same thing
as 1 over n to the fifth

00:01:37.100 --> 00:01:39.670
times r to the fifth.

00:01:39.670 --> 00:01:43.210
Or we could say this is equal
to r to the fifth

00:01:43.210 --> 00:01:47.600
over n to the fifth.

00:01:47.600 --> 00:01:50.200
And this is about as simple
as you can you can get.

00:01:50.200 --> 00:01:52.100
If you wanted to do other things
here you could say that

00:01:52.100 --> 00:01:57.110
this is equal to r over
n to the fifth.

00:01:57.110 --> 00:01:59.710
Since both of these are being
raised to the fifth power.

00:01:59.710 --> 00:02:02.130
But this is probably what
they want here.

00:02:02.130 --> 00:02:05.120
We've simplified it and all of
the exponents are now positive

00:02:05.120 --> 00:02:08.330
because we put the n
in the denominator.


U.S. national debt (scientific notation word problem)

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.890
On February 2, 2010 the U.S.
Treasury estimated the

00:00:04.890 --> 00:00:11.900
national debt at 1.2278 times
10 to the 13th power.

00:00:11.900 --> 00:00:15.930
And just to get a sense of
things, 1 times 10 to the

00:00:15.930 --> 00:00:19.330
sixth is a million, 1 times 10
to the ninth is a billion, 1

00:00:19.330 --> 00:00:21.470
times 10 to the 12th
is a trillion.

00:00:21.470 --> 00:00:23.450
So we're talking on the
order of magnitude

00:00:23.450 --> 00:00:25.210
of 10 trillion dollars.

00:00:25.210 --> 00:00:28.470
So this is about 12
trillion dollars.

00:00:28.470 --> 00:00:31.970
Then they tell us that the U.S.
Census Bureau's estimate

00:00:31.970 --> 00:00:36.390
for the U.S. population was
about 3.086 times 10 to the

00:00:36.390 --> 00:00:37.100
eighth power.

00:00:37.100 --> 00:00:39.875
So this is a little over
300 million people.

00:00:39.875 --> 00:00:41.900
So that's an interesting number
right there, it's the

00:00:41.900 --> 00:00:42.870
population.

00:00:42.870 --> 00:00:46.510
And then they say, using these
estimates calculate the

00:00:46.510 --> 00:00:49.400
per-person share of
the national debt.

00:00:49.400 --> 00:00:51.840
So essentially, we want to
take the entire debt and

00:00:51.840 --> 00:00:53.450
divide by the number
of people.

00:00:53.450 --> 00:00:56.850
That'll give us the per-person
share of the national debt.

00:00:56.850 --> 00:01:00.320
Use scientific notation to make
your calculations and

00:01:00.320 --> 00:01:01.630
express your answer in both

00:01:01.630 --> 00:01:05.710
scientific and decimal notation.

00:01:05.710 --> 00:01:07.860
Which means just as
a regular number.

00:01:07.860 --> 00:01:11.410
Round to four decimal places
while making calculations.

00:01:11.410 --> 00:01:12.990
So we want the per-person
debt.

00:01:12.990 --> 00:01:14.430
So we want to take the total
debt and divide by

00:01:14.430 --> 00:01:16.050
the number of people.

00:01:16.050 --> 00:01:26.270
So the total debt is 1.2278
times 10 to the 13th.

00:01:26.270 --> 00:01:31.610
And we want to divide that by
the total number of people,

00:01:31.610 --> 00:01:37.660
which is 3.086 times
10 to the eighth.

00:01:37.660 --> 00:01:40.160
And we could separate this into
two division problems. We

00:01:40.160 --> 00:01:45.070
could say that this is equal
to the division right here,

00:01:45.070 --> 00:01:56.310
1.2278 divided by 3.086.

00:01:56.310 --> 00:02:02.630
And then times 10 to the 13th
divided by 10 to the eighth.

00:02:02.630 --> 00:02:05.280
Now, what's 10 to the 13th
divided by 10 to the eighth?

00:02:05.280 --> 00:02:06.530
Let me do it over here.

00:02:06.578 --> 00:02:13.590
The way I think about it, this
is the exact same thing as 10

00:02:13.590 --> 00:02:18.860
to the 13th times 10 to
the negative eight.

00:02:18.860 --> 00:02:20.530
This is an eight right here.

00:02:20.530 --> 00:02:22.610
If you have a 10 to the eighth
in the denominator, that's

00:02:22.610 --> 00:02:24.870
like multiplying by 10 to
the negative eight.

00:02:24.870 --> 00:02:28.660
So you have 13, you the same
base 10, so 10 to the 13th

00:02:28.660 --> 00:02:31.770
times 10 to the negative eight
is going to be 10 to

00:02:31.770 --> 00:02:33.390
the 13 minus 8.

00:02:33.390 --> 00:02:35.070
Which is 10 to the fifth.

00:02:35.070 --> 00:02:38.910
Or another way to think about
it: If you have the base in

00:02:38.910 --> 00:02:40.880
the denominator, you subtract
the exponents.

00:02:40.880 --> 00:02:42.190
So it's 13 minus 8.

00:02:42.190 --> 00:02:43.420
10 to the fifth.

00:02:43.420 --> 00:02:52.360
So it's this blue expression
times 10 to the fifth.

00:02:52.360 --> 00:02:57.240
And let's get a calculator out
to calculate this right here.

00:02:57.240 --> 00:02:59.985
And they say round everything
to four decimal places, so

00:02:59.985 --> 00:03:02.550
I'll keep that in mind.

00:03:02.550 --> 00:03:03.720
Let me turn my calculator on.

00:03:03.720 --> 00:03:22.760
1.2278 divided by 3.086
is equal to 0.3979.

00:03:22.760 --> 00:03:26.020
Because we want to round
right there.

00:03:26.020 --> 00:03:27.005
Let me remember that.

00:03:27.005 --> 00:03:30.780
Or let me just put it
on the side so I can

00:03:30.780 --> 00:03:33.020
still look at it.

00:03:33.020 --> 00:03:35.810
So this this little
dividing decimals

00:03:35.810 --> 00:03:44.840
problem results in 0.3979.

00:03:44.840 --> 00:03:51.510
And of course, times 10 to the
fifth dollars per person.

00:03:51.510 --> 00:03:53.640
Once again, you might be tempted
to say, hey this is in

00:03:53.640 --> 00:03:54.590
scientific notation.

00:03:54.590 --> 00:03:57.760
I have some number times
a power of ten.

00:03:57.760 --> 00:04:01.250
But notice, this number is not
greater than or equal to 1.

00:04:01.250 --> 00:04:03.010
Remember, this number, if you
want to be formal about

00:04:03.010 --> 00:04:06.260
scientific notation, has to be
greater than or equal to 1, or

00:04:06.260 --> 00:04:07.770
less than 10.

00:04:07.770 --> 00:04:12.020
So what we can do here
is we can multiply.

00:04:12.020 --> 00:04:14.580
If we don't want to change the
number, we can multiply this

00:04:14.580 --> 00:04:17.630
number by 10 and divide
this number by 10.

00:04:17.630 --> 00:04:20.209
Or another way you can think
about it is, this whole thing

00:04:20.209 --> 00:04:26.180
can be rewritten as
0.3979 times 10

00:04:26.180 --> 00:04:30.110
times 10 to the fourth.

00:04:30.110 --> 00:04:32.510
What I did was just now was I
broke up the 10 of the fifth

00:04:32.510 --> 00:04:35.710
into a 10 and a 10
to the fourth.

00:04:35.710 --> 00:04:38.120
And I did that because I want
to multiply this by 10 so I

00:04:38.120 --> 00:04:41.250
can get a 3 out front
instead of a 0.3.

00:04:41.250 --> 00:04:44.370
So let's do that.

00:04:44.370 --> 00:04:46.400
So essentially, I took a 10 out
of the 10 to the fifth.

00:04:46.400 --> 00:04:48.690
I divided it by 10, I multiplied
this other guy by

00:04:48.690 --> 00:04:50.670
10, not changing the
whole number.

00:04:50.670 --> 00:04:57.840
So then this right here will
become 3.979 and then times 10

00:04:57.840 --> 00:04:59.090
to the fourth.

00:05:03.140 --> 00:05:05.630
So that's how much debt
there is per-person

00:05:05.630 --> 00:05:07.140
in scientific notation.

00:05:07.140 --> 00:05:09.460
So this is debt per person in scientific notation.

00:05:09.460 --> 00:05:15.240
 Now, in the problem they also
wanted us to express it in

00:05:15.240 --> 00:05:17.700
decimal notation.

00:05:17.700 --> 00:05:20.420
Which is just kind of standard
writing it as a number with

00:05:20.420 --> 00:05:23.080
our standard numeric
decimal system.

00:05:23.080 --> 00:05:26.100
So what is 3.979 times
10 to the fourth?

00:05:26.100 --> 00:05:27.350
Let's think about it.

00:05:27.350 --> 00:05:33.490
We have 3.979 times ten to the fourth.

00:05:33.490 --> 00:05:36.830
Well let me just
do it this way.

00:05:36.830 --> 00:05:38.670
Let's just move the
decimal space.

00:05:38.670 --> 00:05:41.910
If we multiply it by 10, we're
going to get 39.79.

00:05:41.910 --> 00:05:43.590
If we multiply it by
10 squared, we're

00:05:43.590 --> 00:05:45.335
going to get 397.9.

00:05:45.335 --> 00:05:47.250
If we multiply it by 10
to the third, we're

00:05:47.250 --> 00:05:49.655
going to get 3,979.

00:05:49.655 --> 00:05:53.020
If we multiply it by 10 to the
fourth, we're going to get one

00:05:53.020 --> 00:05:54.340
more zero right there.

00:05:54.340 --> 00:05:55.320
So we're essentially
going to move the

00:05:55.320 --> 00:05:56.680
decimal four to the right.

00:05:56.738 --> 00:06:00.950
So I could write it like this.

00:06:00.950 --> 00:06:06.620
This is equal to $39,790.

00:06:06.620 --> 00:06:09.000
So if you think about the
national debt per person.

00:06:09.000 --> 00:06:13.070
Every man, woman, and child in
the United States essentially

00:06:13.070 --> 00:06:21.267
owes $39,790.


Scientific notation 2

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.780
The speed of light is
3 times 10 to the

00:00:03.780 --> 00:00:06.520
eighth meters per second.

00:00:06.520 --> 00:00:11.320
If it takes 5 times 10 to the
second, or 5 times 100, or

00:00:11.320 --> 00:00:16.450
however you want to view it,
seconds for light to travel

00:00:16.450 --> 00:00:20.470
from the Sun to the Earth, what
is the distance in meters

00:00:20.470 --> 00:00:21.790
between the Sun and the Earth?

00:00:21.790 --> 00:00:24.510
Express your answer in
scientific notation.

00:00:24.510 --> 00:00:27.150
So this is a simple distance
is equal to

00:00:27.150 --> 00:00:28.440
rate times time problem.

00:00:31.400 --> 00:00:34.690
And I think what they really
want to test is our ability to

00:00:34.690 --> 00:00:37.410
manipulate numbers in
scientific notation.

00:00:37.410 --> 00:00:40.750
So what is our rate here?

00:00:40.750 --> 00:00:43.320
What is the speed of the
thing in question?

00:00:43.320 --> 00:00:47.010
Well they tell us right here,
the speed of light is 3 times

00:00:47.010 --> 00:00:49.590
10 to the eighth meters
per second.

00:00:49.590 --> 00:00:50.840
So we could write that down.

00:00:59.430 --> 00:01:03.000
And then what is the time?

00:01:03.000 --> 00:01:09.760
They say if it takes 5 times 10
to the second seconds for

00:01:09.760 --> 00:01:11.670
light to travel from the
Sun to the Earth,

00:01:11.670 --> 00:01:12.920
so that's the time.

00:01:22.260 --> 00:01:24.330
And they say what's the
distance in meters?

00:01:24.330 --> 00:01:26.680
So I just take the product
of these two things.

00:01:26.680 --> 00:01:29.635
The distance is just going to
be the product of 3 times 10

00:01:29.635 --> 00:01:32.210
to the eighth and 5
times 10 squared.

00:01:32.210 --> 00:01:35.670
Now, what's fun about scientific
notation is that

00:01:35.670 --> 00:01:38.100
you can just rearrange the
numbers you're taking the

00:01:38.100 --> 00:01:40.360
products of, because that's
just the property of

00:01:40.360 --> 00:01:43.340
multiplication, if everything
is just being multiplied,

00:01:43.340 --> 00:01:45.220
order doesn't matter.

00:01:45.220 --> 00:01:53.080
And so we could rewrite this
right here as 3 times 5, I'll

00:01:53.080 --> 00:01:58.320
try to keep the colors the same,
3 times 5 times 10 to

00:01:58.320 --> 00:02:04.670
the eighth times 10 squared.

00:02:04.670 --> 00:02:07.060
And let's see what happens
to the units.

00:02:07.060 --> 00:02:09.949
You have meters per second
times seconds.

00:02:09.949 --> 00:02:12.120
Second in the numerator, second
in the denominator,

00:02:12.120 --> 00:02:13.000
they cancel out.

00:02:13.000 --> 00:02:17.230
And you're left just with
that meters over there.

00:02:17.230 --> 00:02:20.630
And that is nice because our
answer needs to be in meters.

00:02:20.630 --> 00:02:23.080
The distance in meters between
the Sun and the Earth.

00:02:23.080 --> 00:02:24.750
So what's this going to be?

00:02:24.750 --> 00:02:27.370
So you have a 3 times
a 5 right here.

00:02:27.370 --> 00:02:28.710
That's 15.

00:02:28.710 --> 00:02:31.960
And then you have that being
multiplied by 10 to the eighth

00:02:31.960 --> 00:02:33.940
times 10 squared.

00:02:33.940 --> 00:02:36.490
10 to the eighth times
10 squared, that's 10

00:02:36.490 --> 00:02:38.040
to the 8 plus 2.

00:02:38.040 --> 00:02:41.520
Or 10 to the tenth power.

00:02:41.520 --> 00:02:44.470
So the distance is 15 times
10 to the tenth meters.

00:02:44.470 --> 00:02:46.200
Now you might say,
hey, I'm done.

00:02:46.200 --> 00:02:47.710
That is the distance.

00:02:47.710 --> 00:02:49.690
And that looks like scientific
notation.

00:02:49.690 --> 00:02:52.390
I have some number times
some power of 10.

00:02:52.390 --> 00:02:55.010
But remember, if you really want
to be formal about your

00:02:55.010 --> 00:02:58.720
scientific notation, this number
right here has to be

00:02:58.720 --> 00:03:00.030
less then 10.

00:03:00.030 --> 00:03:01.950
The way I've written it right
now, it is clearly

00:03:01.950 --> 00:03:03.030
not less than 10.

00:03:03.030 --> 00:03:04.460
So how can we rewrite it?

00:03:04.460 --> 00:03:05.970
Well what's 15?

00:03:05.970 --> 00:03:13.780
15 is the same thing
as 1.5 times 10.

00:03:13.780 --> 00:03:16.560
So we could write 15
as 1.5 times 10.

00:03:16.560 --> 00:03:23.360
And then it's going to be that
times 10 to the tenth.

00:03:23.360 --> 00:03:25.360
And of course, we have
our units there,

00:03:25.360 --> 00:03:26.820
meters right there.

00:03:26.820 --> 00:03:28.150
And now we're really close.

00:03:28.150 --> 00:03:30.210
We have 1.5 times 10.

00:03:30.210 --> 00:03:34.790
Or we could view that as 1.5
times 10 to the first times 10

00:03:34.790 --> 00:03:35.630
to the tenth.

00:03:35.630 --> 00:03:38.570
And then we have 10 to the first
times 10 to the tenth.

00:03:38.570 --> 00:03:40.460
That is just 10 to
the eleventh.

00:03:43.950 --> 00:03:49.160
So the total distance from
the Sun to the Earth, in

00:03:49.160 --> 00:03:55.550
scientific notation, is 1.5
times these guys, which is 10

00:03:55.550 --> 00:03:58.200
to the eleventh meters.


Exponent Rules 3

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.730
Write in simplified
exponential form.

00:00:03.730 --> 00:00:05.810
And we have a product of
a bunch of things here.

00:00:05.810 --> 00:00:11.980
6 times 7 times 6 times 9 times
2 to the fourth times 2

00:00:11.980 --> 00:00:15.140
times 6 squared times 9.

00:00:15.140 --> 00:00:17.390
So a good place to start is to
just group the things that are

00:00:17.390 --> 00:00:18.360
of the same base.

00:00:18.360 --> 00:00:20.770
And then maybe we can combine
their exponents.

00:00:20.770 --> 00:00:22.720
And let's see if we can simplify
it from there.

00:00:22.720 --> 00:00:25.020
Let's just go in order
of the bases.

00:00:25.020 --> 00:00:26.220
Let's start with the 2's.

00:00:26.220 --> 00:00:28.210
So we have a 2 to
the fourth here.

00:00:28.210 --> 00:00:29.880
And then we have times 2.

00:00:29.880 --> 00:00:32.710
You could view that as times
2 to the first if you like.

00:00:32.710 --> 00:00:33.820
Let me write that there.

00:00:33.820 --> 00:00:36.860
So you have to 2 to the
fourth times 2.

00:00:36.860 --> 00:00:39.960
I'll just rewrite it as 2 to the
first. So we did our 2's.

00:00:39.960 --> 00:00:42.160
The next largest number,
or the next largest

00:00:42.160 --> 00:00:45.450
base, are the 6's.

00:00:45.450 --> 00:00:48.290
We have a 6 right there.

00:00:48.290 --> 00:00:52.040
I'll write it as 6 to the first.
Times another 6 or

00:00:52.040 --> 00:00:54.600
times another 6 to the
first right there.

00:00:54.600 --> 00:00:56.370
Then we have a 6 squared
over here.

00:00:56.370 --> 00:00:58.910
So times 6 squared.

00:00:58.910 --> 00:01:02.470
I'm really just rearranging
this product out here.

00:01:02.470 --> 00:01:05.360
Then the next number that we
have here, the next largest

00:01:05.360 --> 00:01:06.190
one, we have a 7.

00:01:06.190 --> 00:01:07.390
We only have one 7 there.

00:01:07.390 --> 00:01:10.580
So I'll just write times 7.

00:01:10.580 --> 00:01:13.170
And then finally, we have
a couple of 9's.

00:01:17.700 --> 00:01:20.800
I'll write it as 9 to the one,
then we have another 9.

00:01:20.800 --> 00:01:23.040
Times 9 to the one, or
we could just write

00:01:23.040 --> 00:01:25.080
that as 9 times 9.

00:01:25.080 --> 00:01:27.000
Now, let's merge the bases.

00:01:27.000 --> 00:01:30.400
Or let's merge the products
of these bases raised to

00:01:30.400 --> 00:01:31.680
different exponents.

00:01:31.680 --> 00:01:36.550
You have 2 to the fourth power
times 2 to the first power.

00:01:36.550 --> 00:01:38.510
This is going to be the
same thing as 2 to

00:01:38.510 --> 00:01:40.140
the 4 plus 1 power.

00:01:40.140 --> 00:01:42.790
Or 2 to the fifth power.

00:01:42.790 --> 00:01:45.460
Then you have your 6's.

00:01:45.460 --> 00:01:48.530
So 6 to the first times 6 to the
first times 6 squared is

00:01:48.530 --> 00:01:53.960
the same thing as 6 to
that 1 plus 1 plus 2.

00:01:53.960 --> 00:01:56.370
Or 6 to the fourth power.

00:01:56.370 --> 00:01:57.590
Then we have that 7 there.

00:01:57.590 --> 00:01:59.840
Nothing really you can
do to simplify that.

00:01:59.840 --> 00:02:01.740
Times that 7.

00:02:01.740 --> 00:02:04.710
And then we have a 9 times
a 9, which is 9 squared.

00:02:04.710 --> 00:02:07.020
Or you can think about it as 9
to the first times 9 to the

00:02:07.020 --> 00:02:08.865
first, add the exponents,
9 squared.

00:02:12.190 --> 00:02:14.810
So depending on how you want to
do this problem, the people

00:02:14.810 --> 00:02:16.780
who actually wrote this problem,
this is what they

00:02:16.780 --> 00:02:20.610
wanted, they viewed this as
simplified exponential form.

00:02:20.610 --> 00:02:24.730
We've combined the same bases
raised to different powers.

00:02:24.730 --> 00:02:26.680
So you could stop here.

00:02:26.680 --> 00:02:30.180
But if you really want to get
it down to kind of the real

00:02:30.180 --> 00:02:31.400
simplest form.

00:02:31.400 --> 00:02:34.500
What you could essentially do
is break everything down to

00:02:34.500 --> 00:02:35.840
their prime factors.

00:02:35.840 --> 00:02:37.840
And then have those raised
to exponents and then

00:02:37.840 --> 00:02:39.550
merge them even more.

00:02:39.550 --> 00:02:41.260
Let me show you what
I'm talking about.

00:02:41.260 --> 00:02:43.020
So this 6 to the fourth here.

00:02:43.020 --> 00:02:44.950
Let me keep it the same color.

00:02:44.950 --> 00:02:48.150
So we have the 2 to the fifth,
let me just write that there.

00:02:48.150 --> 00:02:55.710
But the 6 to the fourth, I could
rewrite 6 as 2 times 3

00:02:55.710 --> 00:02:56.920
to the fourth.

00:02:56.920 --> 00:02:59.710
6 to the fourth is the exact
same thing is that.

00:02:59.710 --> 00:03:02.570
Then we have the times 7.

00:03:02.570 --> 00:03:05.210
Then we have the 9 squared.

00:03:05.210 --> 00:03:10.420
But 9 we can rewrite
as 3 squared

00:03:10.420 --> 00:03:12.070
and all of that squared.

00:03:12.070 --> 00:03:16.290
All did here is I rewrote the
6 as 2 times 3 and I rewrote

00:03:16.290 --> 00:03:20.640
the 9 as 3 times 3
or as 3 squared.

00:03:20.640 --> 00:03:22.970
I want to keep the color
codes the same way.

00:03:22.970 --> 00:03:26.000
So we had the 2 to the fifth.

00:03:26.000 --> 00:03:29.220
Now, what's 2 times
3 to the fourth?

00:03:29.220 --> 00:03:30.590
Well here's an exponent
property.

00:03:30.590 --> 00:03:33.430
If you take the product of two
things and it's to a power,

00:03:33.430 --> 00:03:37.410
that's the same thing
as 2 to the fourth

00:03:37.410 --> 00:03:39.890
times 3 to the fourth.

00:03:39.890 --> 00:03:43.990
And then you have this 7 here.

00:03:43.990 --> 00:03:45.480
So I essentially just
raised both of

00:03:45.480 --> 00:03:46.550
these guys to the fourth.

00:03:46.550 --> 00:03:49.220
Because their product is being
raised to the fourth power.

00:03:49.220 --> 00:03:52.800
And then you have times 3
squared, but you have 3

00:03:52.800 --> 00:03:54.620
squared and then that squared.

00:03:54.620 --> 00:03:57.670
So it's going to be 3 to
the 2 times 2 power.

00:03:57.670 --> 00:04:00.490
So times 3 to the fourth.

00:04:00.490 --> 00:04:02.570
So now we have more bases
we can merge.

00:04:02.570 --> 00:04:06.270
We have these 2's here.

00:04:06.270 --> 00:04:09.090
2 to the fifth times 2 to
the fourth, what's that?

00:04:09.090 --> 00:04:11.620
Well that's 2 to the 5 plus 4.

00:04:11.620 --> 00:04:13.660
That's 2 to the ninth power.

00:04:13.660 --> 00:04:15.340
Then you have your
3's over here.

00:04:15.340 --> 00:04:17.990
Let me do that in green.

00:04:17.990 --> 00:04:21.180
You have 3 to the fourth and 3
to the fourth, what's that

00:04:21.180 --> 00:04:22.050
going to be?

00:04:22.050 --> 00:04:25.960
That's going to be times 3 to
the 4 plus fourth power, or 3

00:04:25.960 --> 00:04:27.440
to the eighth power.

00:04:27.440 --> 00:04:31.010
And then finally, you have this
7 sitting over there.

00:04:31.010 --> 00:04:33.190
So if you really wanted to break
things down to their

00:04:33.190 --> 00:04:37.270
prime factors and kind of have
the the least bases here, you

00:04:37.270 --> 00:04:40.300
can view this is even a more
simplified exponential form

00:04:40.300 --> 00:04:44.310
than what the question writers
originally wanted.

00:04:44.310 --> 00:04:47.170
So you could get down all the
way to 2 to the ninth times 3

00:04:47.170 --> 00:04:49.350
to the eighth times 7, which
is the same thing is that,

00:04:49.350 --> 00:04:52.270
which is the exact same
thing as that.


Graphing systems of inequalities 3

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:04.590
Luis receives a gift card
worth $25 to an online

00:00:04.590 --> 00:00:07.560
retailer the sells digital
music and games.

00:00:07.560 --> 00:00:12.440
Each song costs $0.89 and
each game costs $1.99.

00:00:12.440 --> 00:00:15.580
He wants to buy at least 15
items with this card.

00:00:15.580 --> 00:00:18.140
And then they want us to set
up a system of inequalities

00:00:18.140 --> 00:00:21.410
that represents the scenario and
then identify the range of

00:00:21.410 --> 00:00:24.505
possible purchases
using a graph.

00:00:24.505 --> 00:00:27.710
So let's see if we
can do this.

00:00:27.710 --> 00:00:30.410
So he's going to buy some
amount of games and some

00:00:30.410 --> 00:00:31.550
amount of music.

00:00:31.550 --> 00:00:35.670
So let's make those
our two variables.

00:00:35.670 --> 00:00:40.680
Let's let x equal the
number of music, or

00:00:40.680 --> 00:00:42.410
the amount of music.

00:00:42.410 --> 00:00:46.060
And let's let y equal the amount
of games, or the number

00:00:46.060 --> 00:00:50.250
of game units or whatever
you want to call it.

00:00:50.250 --> 00:00:52.420
So we have the first constraint
here that the gift

00:00:52.420 --> 00:00:54.020
card is worth $25.

00:00:54.020 --> 00:00:57.810
So I'll underline, the gift
card is worth $25.

00:00:57.810 --> 00:01:00.680
So the total amount he spends
on games and music combined

00:01:00.680 --> 00:01:02.470
can be no more than $25.

00:01:02.470 --> 00:01:04.660
So how much is he going
to spend on music?

00:01:04.660 --> 00:01:06.770
Well each song costs $0.89.

00:01:06.770 --> 00:01:10.910
So he's going to spend $0.89
times the number of songs he

00:01:10.910 --> 00:01:12.200
buys on music.

00:01:12.200 --> 00:01:14.130
Plus how much is he going
to spend on games?

00:01:14.130 --> 00:01:18.620
Well each game is $1.99.

00:01:18.620 --> 00:01:21.250
So he's going to spend
$1.99 times the

00:01:21.250 --> 00:01:23.690
number of games he buys.

00:01:23.690 --> 00:01:29.520
And that has to be less
than or equal to 25.

00:01:29.520 --> 00:01:31.010
So that's our first
constraint.

00:01:31.010 --> 00:01:33.550
And then the second constraint,
is that he wants

00:01:33.550 --> 00:01:36.740
to buy at least 15 items
with his card.

00:01:36.740 --> 00:01:39.410
So the number of songs plus the
number of games has to be

00:01:39.410 --> 00:01:40.880
at least 15.

00:01:40.880 --> 00:01:46.950
so x plus y has to be a greater
than or equal to 15.

00:01:46.950 --> 00:01:49.170
And so just like that we have
our two constraints.

00:01:49.170 --> 00:01:51.670
We've set up the system of
inequalities that represents

00:01:51.670 --> 00:01:52.580
the scenario.

00:01:52.580 --> 00:01:56.180
Now we need to identify the
range of possible purchases

00:01:56.180 --> 00:01:57.430
using a graph.

00:01:57.430 --> 00:01:59.400
And this is a pretty
straightforward graph to draw,

00:01:59.400 --> 00:02:00.160
this one down here.

00:02:00.160 --> 00:02:01.850
This one is the little less
straightforward, because it

00:02:01.850 --> 00:02:04.210
doesn't have nice
round numbers.

00:02:04.210 --> 00:02:07.900
We could round it off to say
this is 1x plus 2y, but

00:02:07.900 --> 00:02:09.690
instead of doing that, let's
just take out a graphing

00:02:09.690 --> 00:02:12.020
calculator and graph
it on that.

00:02:12.020 --> 00:02:15.960
And actually before I graph it,
let me put it in y equals

00:02:15.960 --> 00:02:18.640
mx plus b form, or slope
intercept form.

00:02:18.640 --> 00:02:21.220
So the second one right here,
if we subtract x from both

00:02:21.220 --> 00:02:24.030
sides, we get y is greater
than or equal to

00:02:24.030 --> 00:02:26.300
negative x plus 15.

00:02:26.300 --> 00:02:28.000
So that's the second equation.

00:02:28.000 --> 00:02:32.650
This first equation, if we
subtract 0.89x from both

00:02:32.650 --> 00:02:47.580
sides, we get 1.99y is equal
to negative 0.89x plus 25.

00:02:47.580 --> 00:02:50.370
Sorry, that should be less than
or equal to, not equal

00:02:50.370 --> 00:02:51.990
to, this is an inequality.

00:02:51.990 --> 00:02:55.170
So negative 0.89x plus 25.

00:02:55.170 --> 00:03:04.980
And then divide both
sides by 1.99.

00:03:04.980 --> 00:03:09.500
And we get y is less than or
equal to, the calculator will

00:03:09.500 --> 00:03:18.535
actually be useful for this, we
have negative 0.89 divided

00:03:18.535 --> 00:03:29.110
by 1.99 is equal to negative
0.447 I think is pretty good.

00:03:29.110 --> 00:03:36.380
Negative 0.447x plus
25 divided by 1.99.

00:03:36.380 --> 00:03:45.280
So 25 divided by 1.99
is equal to 12.56.

00:03:45.280 --> 00:03:48.540
Plus 12.56.

00:03:48.540 --> 00:03:50.970
So that that blue constraint,
that we can't spend more than

00:03:50.970 --> 00:03:54.170
$25, when we put it in slope
intercept, it turns into this

00:03:54.170 --> 00:03:54.860
inequality.

00:03:54.860 --> 00:03:56.810
And the second constraint, is
that he wants to buy at least

00:03:56.810 --> 00:03:59.110
15 items, turns into
this inequality.

00:03:59.110 --> 00:04:01.060
So let's graph the
two inqualities.

00:04:01.060 --> 00:04:03.250
Or let's graph the lines
and think about

00:04:03.250 --> 00:04:05.330
the region of overlap.

00:04:05.330 --> 00:04:07.980
So if I have my calculator out,
I should have it in a

00:04:07.980 --> 00:04:10.240
place where I can I look
at both graphs.

00:04:10.240 --> 00:04:13.800
So let me to click on graph.

00:04:13.800 --> 00:04:19.450
So this first one is
negative x plus 15.

00:04:19.450 --> 00:04:21.220
And then press enter.

00:04:21.220 --> 00:04:33.610
The second equation is negative
0.447x plus 12.56.

00:04:33.610 --> 00:04:35.170
And now we're ready
to graph them.

00:04:38.550 --> 00:04:41.150
And let's see what happens.

00:04:41.150 --> 00:04:43.930
OK, so it's it looks like we're
a little bit out of the

00:04:43.930 --> 00:04:46.970
range of where you would expect
to be hanging out here.

00:04:46.970 --> 00:04:48.530
So let me change the range.

00:04:48.530 --> 00:04:50.130
Let me go into the
positive range.

00:04:50.130 --> 00:04:53.640
Which makes sense, because he
can only buy positive units of

00:04:53.640 --> 00:04:54.710
things to begin with.

00:04:54.710 --> 00:04:57.890
Let's make the minimum
x-value here zero.

00:04:57.890 --> 00:05:03.060
Let's make the maximum
x-value here 20.

00:05:03.060 --> 00:05:05.750
Let's make the x-scale 1.

00:05:05.750 --> 00:05:07.260
The minimum y-value is 0.

00:05:07.260 --> 00:05:11.070
He can't buy a negative
number of games.

00:05:11.070 --> 00:05:14.110
Let's make y-max 20 and
y-scale is 1 and

00:05:14.110 --> 00:05:15.360
now let's graph it.

00:05:18.080 --> 00:05:18.590
There you go.

00:05:18.590 --> 00:05:20.990
So we're sitting in the
positive quandrant.

00:05:20.990 --> 00:05:27.190
So the first graph, this one on
the top right here, this is

00:05:27.190 --> 00:05:28.140
the orange graph.

00:05:28.140 --> 00:05:33.320
This is y is greater than or
equal to negative x plus 15.

00:05:33.320 --> 00:05:36.190
And actually, even better, so
I can color it in, I'm going

00:05:36.190 --> 00:05:38.490
to pause the video for a second
and try to copy and

00:05:38.490 --> 00:05:40.120
paste this image so
I can color it in.

00:05:40.120 --> 00:05:42.050
Hold on one second.

00:05:42.050 --> 00:05:45.200
So I've pasted it in here so
I can actually draw on it.

00:05:45.200 --> 00:05:47.610
So this first graph here, y is
greater than or equal to

00:05:47.610 --> 00:05:50.560
negative x plus 15, this is this
top line, because it has

00:05:50.560 --> 00:05:52.450
a higher y-intercept than
this other guy.

00:05:52.450 --> 00:05:54.060
This guy is 12.6.

00:05:54.060 --> 00:05:57.460
So let me color it in
that same color.

00:05:57.460 --> 00:06:01.610
So it is that line
right there.

00:06:01.610 --> 00:06:04.340
You can see it has slope
of negative 1.

00:06:04.340 --> 00:06:05.085
So it's that line.

00:06:05.085 --> 00:06:07.750
And we want to be greater than
or equal to that line.

00:06:07.750 --> 00:06:10.430
So we want all of the values
of above that line,

00:06:10.430 --> 00:06:11.360
including that line.

00:06:11.360 --> 00:06:13.020
We could also be equal
to that line.

00:06:13.020 --> 00:06:15.250
So it's everything
above that line.

00:06:15.250 --> 00:06:17.990
Now this blue line, this is the
other one, it's this one

00:06:17.990 --> 00:06:19.240
right here.

00:06:23.380 --> 00:06:25.630
And we have to be less
than or equal to it.

00:06:25.630 --> 00:06:28.750
And we can see it has
a slope less than 1.

00:06:28.750 --> 00:06:32.310
Or a slope less negative
than 1.

00:06:32.310 --> 00:06:33.960
It's actually a larger number.

00:06:33.960 --> 00:06:37.000
But it's a less downward
sloping, and that's because

00:06:37.000 --> 00:06:39.375
negative 0.44 as opposed
to negative 1.

00:06:39.375 --> 00:06:41.110
And it has a lower
y-intercept.

00:06:41.110 --> 00:06:42.820
And we want all the
values below it or

00:06:42.820 --> 00:06:43.800
equal to that line.

00:06:43.800 --> 00:06:46.540
So all of these values
over here.

00:06:46.540 --> 00:06:50.280
Now, the area that overlaps
between these two constraints,

00:06:50.280 --> 00:06:53.260
I'll do it in this yellow color,
is below the blue line,

00:06:53.260 --> 00:06:56.220
including the blue line, and
above the red line, including

00:06:56.220 --> 00:06:56.800
the red light.

00:06:56.800 --> 00:06:58.760
So it's this area right here.

00:07:03.050 --> 00:07:10.770
So any combination of games
and music that are in that

00:07:10.770 --> 00:07:15.010
area will meet both of
his constraints.

00:07:15.010 --> 00:07:17.160
And one thing you might want to
consider, since you can't

00:07:17.160 --> 00:07:22.090
buy half a game or half a song,
it would only be kind of

00:07:22.090 --> 00:07:25.630
the integer coordinates in this
region that would satisfy

00:07:25.630 --> 00:07:26.490
his constraints.

00:07:26.490 --> 00:07:32.230
So however many games, however
many songs this is and music

00:07:32.230 --> 00:07:35.250
is, this would satisfy
his constraints.


Graphing systems of inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:03.640
We're asked to graph the
solution set for this system.

00:00:03.640 --> 00:00:06.260
So we have two inequalities
here, so this is a system of

00:00:06.260 --> 00:00:09.750
inequalities, and we know how
to graph them independently.

00:00:09.750 --> 00:00:13.430
And so when you graph the
system, you want to find the

00:00:13.430 --> 00:00:17.030
set of coordinates in the
coordinate plane that satisfy

00:00:17.030 --> 00:00:18.940
both of these inequalities.

00:00:18.940 --> 00:00:21.290
So we essentially want to
find out where these two

00:00:21.290 --> 00:00:22.845
inequalities overlap.

00:00:22.845 --> 00:00:25.450
So let's graph this first
one over here.

00:00:25.450 --> 00:00:26.830
I'll rewrite it in magenta.

00:00:26.830 --> 00:00:30.400
y is greater than x minus 8.

00:00:30.400 --> 00:00:33.290
So the way to think about this,
let's think about y is

00:00:33.290 --> 00:00:36.020
equal x minus 8, but we're going
to be greater than that,

00:00:36.020 --> 00:00:38.130
so we're going to draw
a dotted line at y is

00:00:38.130 --> 00:00:39.880
equal x minus 8.

00:00:39.880 --> 00:00:42.970
So if you think about x minus 8,
the y-intercept is negative

00:00:42.970 --> 00:00:49.260
8, so it's one, two, three,
four, five, six, seven, eight.

00:00:49.260 --> 00:00:50.890
That is negative
8 right there.

00:00:50.890 --> 00:00:52.880
This, of course,
is the y-axis.

00:00:52.880 --> 00:00:54.360
That is the x-axis.

00:00:54.360 --> 00:00:57.500
And the graph of y equals
x minus 8 has

00:00:57.500 --> 00:00:59.090
a slope of 1, right?

00:00:59.090 --> 00:01:01.190
There's 1 times this
x right here.

00:01:01.190 --> 00:01:03.420
So every time you move in the
positive x-direction, you move

00:01:03.420 --> 00:01:05.730
1 in the positive y-direction.

00:01:05.730 --> 00:01:08.170
If you move 2, you get 2 in
the positive y-direction.

00:01:08.170 --> 00:01:10.290
If you moved 3, you would
go up 3 in the positive

00:01:10.290 --> 00:01:11.300
y-direction.

00:01:11.300 --> 00:01:13.530
And likewise, if you went in
the negative x-direction.

00:01:13.530 --> 00:01:16.180
You'd also go in the negative
y-direction by 1.

00:01:16.180 --> 00:01:20.810
So this line, if you said y is
equal to x minus 8, it would

00:01:20.810 --> 00:01:22.170
be a line just like that.

00:01:22.170 --> 00:01:24.330
But since we're not going to
include it in this set, we

00:01:24.330 --> 00:01:26.100
have to be greater
than that line.

00:01:26.100 --> 00:01:28.072
I'll do a dotted line here.

00:01:28.072 --> 00:01:31.930
Let me do my best to draw
a dotted line, which is

00:01:31.930 --> 00:01:34.980
surprisingly easier to draw
that a non-dotted line.

00:01:34.980 --> 00:01:38.640
So that is essentially a dotted
line version of y is

00:01:38.640 --> 00:01:40.030
equal to x minus 8.

00:01:40.030 --> 00:01:41.290
But we're going to be
greater than that.

00:01:41.290 --> 00:01:45.050
So for any x, x minus 8 is going
to be on that line, but

00:01:45.050 --> 00:01:47.560
the y is going to be all the
values greater than that.

00:01:47.560 --> 00:01:50.115
So you pick any x, this is x
minus 8, the y-value is going

00:01:50.115 --> 00:01:52.910
to be all the y's greater
than that for that x.

00:01:52.910 --> 00:01:56.850
So the solution set for y is
greater than x minus 8 is all

00:01:56.850 --> 00:01:59.750
of the area above this line
and it doesn't include it.

00:01:59.750 --> 00:02:01.370
And that's why I put a
dotted line there.

00:02:01.370 --> 00:02:03.260
So let me do that in
a lighter color.

00:02:03.260 --> 00:02:08.780
So literally, all of this area
up here is the solution set

00:02:08.780 --> 00:02:12.820
for this first inequality.

00:02:12.820 --> 00:02:14.170
Now, let's do the second
inequality.

00:02:14.170 --> 00:02:16.150
I'll do it in another
dark color.

00:02:16.150 --> 00:02:20.550
We have y is less than 5 minus
x, or we could write that as

00:02:20.550 --> 00:02:23.360
negative x plus 5, so it looks
a little bit more like our

00:02:23.360 --> 00:02:25.240
slope-intercept form.

00:02:25.240 --> 00:02:29.530
Now, if we were just to graph y
is equal to negative x plus

00:02:29.530 --> 00:02:31.210
5, it would look like this.

00:02:31.210 --> 00:02:34.910
We would go in the positive
y-direction one, two, three,

00:02:34.910 --> 00:02:36.100
four, five.

00:02:36.100 --> 00:02:38.560
That is the y-intercept
right there.

00:02:38.560 --> 00:02:40.710
And it has a slope
of negative 1.

00:02:40.710 --> 00:02:43.180
So if we move 1 in the
x-direction, we move down 1.

00:02:43.180 --> 00:02:44.900
It we move 3 in the x-direction,
we're going to

00:02:44.900 --> 00:02:46.030
move down 3.

00:02:46.030 --> 00:02:47.650
If we move back 3 in the
x-direction, we're

00:02:47.650 --> 00:02:49.320
going to move up 3.

00:02:49.320 --> 00:02:51.280
So this line is going
to look like this.

00:02:51.280 --> 00:02:53.080
And I'm to do it in a dotted
line because we're not

00:02:53.080 --> 00:02:53.600
including it.

00:02:53.600 --> 00:02:54.760
It's not less than or equal.

00:02:54.760 --> 00:02:57.960
It's just strictly less than, so
it won't include the line.

00:02:57.960 --> 00:03:00.875
So it's going to look
something like this.

00:03:00.875 --> 00:03:06.491
I'll try my best to draw it
as neatly as possible.

00:03:06.491 --> 00:03:09.810
I think that is a
respectable job.

00:03:09.810 --> 00:03:11.960
This is the dotted line version
of y is equal to

00:03:11.960 --> 00:03:13.430
negative x plus 5.

00:03:13.430 --> 00:03:15.320
It's dotted because we're
not including it.

00:03:15.320 --> 00:03:18.800
Now, for any x, you pick an x,
let's say x is equal to one,

00:03:18.800 --> 00:03:21.250
negative x plus 5 is going
to get on to that line.

00:03:21.250 --> 00:03:23.870
But for that x, we only care
about the y's that are less

00:03:23.870 --> 00:03:25.110
than that value.

00:03:25.110 --> 00:03:27.970
So it's going to be all the
values below this line.

00:03:27.970 --> 00:03:30.950
So if you look at the solution
set for this second

00:03:30.950 --> 00:03:34.900
inequality, it will
look like this.

00:03:34.900 --> 00:03:38.360
I'll do it in this
light blue color.

00:03:38.360 --> 00:03:41.160
It will be all of the area
below this blue line.

00:03:47.980 --> 00:03:52.400
Now, what we care about is the
set of coordinates that

00:03:52.400 --> 00:03:54.960
satisfy both of these
inequalities.

00:03:54.960 --> 00:03:57.180
Now, in order to satisfy both
inequalities, it has to be the

00:03:57.180 --> 00:03:59.140
overlap of the solution sets.

00:03:59.140 --> 00:04:01.590
And the overlap of the solution
sets, where the blue

00:04:01.590 --> 00:04:04.930
is overlapping with the yellow
in this case, is everything

00:04:04.930 --> 00:04:08.230
that is above this line and
everything that's below that

00:04:08.230 --> 00:04:09.840
line, so it's all of
this area here.

00:04:09.840 --> 00:04:12.250
So let me color that in
a little bit bolder.

00:04:12.250 --> 00:04:14.160
So it's all of this below
the blue line and

00:04:14.160 --> 00:04:15.250
above the red line.

00:04:15.250 --> 00:04:16.250
This is the overlap.

00:04:16.250 --> 00:04:21.570
All of this area, all of that
area right there is the

00:04:21.570 --> 00:04:24.610
solution set for this system.

00:04:24.610 --> 00:04:25.670
And you can verify it.

00:04:25.670 --> 00:04:29.140
Pick a random point in this
area and see it'll satisfy

00:04:29.140 --> 00:04:29.670
both points.

00:04:29.670 --> 00:04:33.260
And a really easy one is x is
equal to 0, y is equal to 0.

00:04:33.260 --> 00:04:34.880
That's part of our
solution set.

00:04:34.880 --> 00:04:36.340
Let's try it out.

00:04:36.340 --> 00:04:41.250
If y is 0, 0 is greater
than 0 minus 8.

00:04:41.250 --> 00:04:41.710
That's true.

00:04:41.710 --> 00:04:43.380
0 is greater than negative 8.

00:04:43.380 --> 00:04:47.410
If you do the second equation,
0 is less than 5 minus 0.

00:04:47.410 --> 00:04:47.970
That's true.

00:04:47.970 --> 00:04:50.160
0 is less than 5.

00:04:50.160 --> 00:04:51.810
And you could try that
with any points here.

00:04:51.810 --> 00:04:53.940
This orange area is
our solution set.


Mixture problems 3

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:02.770
Make a table and solve.

00:00:02.770 --> 00:00:06.060
A biologist is researching the
impact of three different

00:00:06.060 --> 00:00:09.690
water-based sugar drinks on bees
ability to make honey.

00:00:09.690 --> 00:00:15.540
He takes 2 liters of Drink A,
which contains 40% sugar.

00:00:15.540 --> 00:00:16.490
So let me write this down.

00:00:16.490 --> 00:00:18.900
Let me make our table and
then we can solve it.

00:00:18.900 --> 00:00:27.560
So let's take amount of drink.

00:00:27.560 --> 00:00:29.190
And then we'll say
percent sugar.

00:00:32.479 --> 00:00:35.760
And then we can say sugar
quantity, so the actual

00:00:35.760 --> 00:00:37.170
physical quantity of sugar.

00:00:40.180 --> 00:00:42.490
Maybe I should say sugar amount,
or amount of sugar.

00:00:47.920 --> 00:00:52.800
Now this first drink, Drink A,
it says he takes 2 liters of

00:00:52.800 --> 00:00:59.110
Drink A, which contains
40% sugar.

00:00:59.110 --> 00:01:01.390
The first column will be which
drink we're talking about, so

00:01:01.390 --> 00:01:04.160
Drink A, he takes
2 liters of it.

00:01:08.730 --> 00:01:12.040
It's 40% sugar.

00:01:12.040 --> 00:01:14.750
So if we want the actual amount
of sugar in liters, we

00:01:14.750 --> 00:01:21.220
just multiply 2 liters times
40%, or times 0.4.

00:01:21.220 --> 00:01:24.080
Let me write times with a dot so
you don't think it's an x.

00:01:24.080 --> 00:01:31.610
2 times 0.4, which is equal
to 0.8 liters of sugar.

00:01:31.610 --> 00:01:33.580
So you have 0.8 liters
of sugar.

00:01:33.580 --> 00:01:36.810
1.2 liters of I guess the other
stuff in there is water.

00:01:36.810 --> 00:01:41.350
But it's 0.8 of the 2 liters
is sugar, which is 40%.

00:01:41.350 --> 00:01:45.240
Now,, he adds 1.2 liters
of Drink B.

00:01:55.070 --> 00:01:58.860
He finds that bees prefer this
new solution, Drink C.

00:01:58.860 --> 00:02:01.150
So when you add these two
together, you end

00:02:01.150 --> 00:02:02.400
up with Drink C.

00:02:04.730 --> 00:02:06.350
And we end up with how
much of Drink C?

00:02:06.350 --> 00:02:12.770
2 plus 1.2 is 3.2 liters
of Drink C, which

00:02:12.770 --> 00:02:15.420
has 25% sugar content.

00:02:15.420 --> 00:02:22.110
So this is 25% sugar, which also
says we know the amount

00:02:22.110 --> 00:02:23.170
of sugar in it.

00:02:23.170 --> 00:02:27.600
Because if we have 3.2 liters
of it and it's 25% sugar, or

00:02:27.600 --> 00:02:31.970
it's 1/4 sugar, that means that
we have 0.8 liters of

00:02:31.970 --> 00:02:32.670
sugar here.

00:02:32.670 --> 00:02:39.380
So this is 0.8 liters
of sugar.

00:02:39.380 --> 00:02:41.060
Well, that I already wrote
in the column name.

00:02:41.060 --> 00:02:42.090
That's the amount of sugar.

00:02:42.090 --> 00:02:43.550
It's 25% sugar.

00:02:43.550 --> 00:02:46.350
We have 3.2 liters of it.

00:02:46.350 --> 00:02:48.760
Now, they want to know what
is the percentage of

00:02:48.760 --> 00:02:50.580
sugar in Drink B?

00:02:50.580 --> 00:02:51.990
So let's just call that x.

00:02:56.030 --> 00:02:58.690
So that's right over here.

00:02:58.690 --> 00:03:03.030
Now, if it's x percent sugar
here, or this is the decimal

00:03:03.030 --> 00:03:05.970
equivalent, that's x, how
much sugar do we have?

00:03:05.970 --> 00:03:09.770
We have 1.2 liters times the
decimal equivalent of sugar,

00:03:09.770 --> 00:03:12.660
so this is going to
be 1.2 times x.

00:03:15.720 --> 00:03:16.790
Now let's think about it.

00:03:16.790 --> 00:03:22.560
We have 0.8 liters of sugar in
Drink A, and when you add this

00:03:22.560 --> 00:03:26.550
amount to it, you still have
0.8 liters of actual

00:03:26.550 --> 00:03:28.080
sugar in Drink C.

00:03:28.080 --> 00:03:29.980
So this thing has to
be equal to zero.

00:03:29.980 --> 00:03:31.250
We could set up an
equation here.

00:03:31.250 --> 00:03:42.770
We could write 0.8 plus
1.2x is equal to 0.8.

00:03:42.770 --> 00:03:44.930
You subtract 0.8 from
both sides.

00:03:44.930 --> 00:03:50.000
You get 1.2x is equal to 0. x
has got to be equal to 0.

00:03:50.000 --> 00:03:52.570
So this thing right here
has got to be zero.

00:03:52.570 --> 00:03:55.280
There's no sugar in Drink B.

00:03:55.280 --> 00:03:57.250
It's just got to be
like 1.2 liters.

00:03:57.250 --> 00:03:59.170
I guess the solution is water.

00:03:59.170 --> 00:04:01.420
So it's 1.2 liters of water.

00:04:01.420 --> 00:04:03.670
There's no sugar in Drink B.

00:04:03.670 --> 00:04:06.080
It is 0% sugar.