Fractions cut and copy 2 exercise example

WEBVTT
Kind: captions
Language: en

00:00:00.910 --> 00:00:03.770
The starting block
below is 2 units long.

00:00:03.770 --> 00:00:06.500
Use the tools to the right
to convert the starting block

00:00:06.500 --> 00:00:08.290
into both of the
goal blocks, that

00:00:08.290 --> 00:00:12.680
are 5/8 units long, and
1 and 3/4 units long.

00:00:12.680 --> 00:00:14.754
You can see your progress
as the current block.

00:00:14.754 --> 00:00:16.170
So what are they
telling us to do?

00:00:16.170 --> 00:00:18.460
So this is the starting
block, right over here.

00:00:18.460 --> 00:00:20.380
They're telling us
this is 2 units long.

00:00:20.380 --> 00:00:22.690
So a unit must be about
this long, right here.

00:00:22.690 --> 00:00:24.810
About half of this
starting block.

00:00:24.810 --> 00:00:26.560
And we have these
tools on the right.

00:00:26.560 --> 00:00:28.800
And my first goal block,
right over here in blue,

00:00:28.800 --> 00:00:30.230
they say that that's 5/8.

00:00:30.230 --> 00:00:33.610
And that makes sense, because
if this is about a whole,

00:00:33.610 --> 00:00:35.170
then that looks like about 5/8.

00:00:35.170 --> 00:00:37.130
This right over here
would be about 8/8.

00:00:37.130 --> 00:00:38.590
So this looks about 5/8.

00:00:38.590 --> 00:00:40.590
And then they tell us
that this other one that's

00:00:40.590 --> 00:00:43.450
below my screen right
over here is 1 and 3/4.

00:00:43.450 --> 00:00:45.990
So they tell us that this
is 1 and 3/4 units long.

00:00:45.990 --> 00:00:48.354
1 and 3/4 is also the
same thing as 7/4.

00:00:48.354 --> 00:00:50.520
So I somehow need to-- let's
see, what are my tools?

00:00:50.520 --> 00:00:52.840
What are the tools
available to me?

00:00:52.840 --> 00:00:56.240
Well I can cut my starting
block into a certain number

00:00:56.240 --> 00:00:57.170
of pieces.

00:00:57.170 --> 00:01:01.200
And then I'm going to copy
that starting block over here

00:01:01.200 --> 00:01:02.830
for block 1, and
I'm going to try

00:01:02.830 --> 00:01:04.379
to match it to this goal block.

00:01:04.379 --> 00:01:06.410
And then I'm also going
to copy it for block 2,

00:01:06.410 --> 00:01:08.720
to match that second goal block.

00:01:08.720 --> 00:01:12.470
So I'm going to try to
get something that's 5/8,

00:01:12.470 --> 00:01:16.140
and I'm also going to try to
get something that's 1 and 3/4.

00:01:16.140 --> 00:01:18.037
Now 3/4 is the
same thing as 6/8.

00:01:18.037 --> 00:01:20.120
And the reason why I care
about eighths is eighths

00:01:20.120 --> 00:01:21.190
is a smaller unit.

00:01:21.190 --> 00:01:25.540
So if I can convert the
starting block into eighths,

00:01:25.540 --> 00:01:28.130
then I can copy it enough
times to get exactly

00:01:28.130 --> 00:01:31.120
to 5/8, and exactly
to 1 and 3/4,

00:01:31.120 --> 00:01:35.660
which is the same thing as-- 3/4
is the exact same thing as 6/8.

00:01:35.660 --> 00:01:38.430
So let's get our starting
block into eighths.

00:01:38.430 --> 00:01:42.300
So if we're starting
with 2 wholes,

00:01:42.300 --> 00:01:46.040
if we're starting with 2 units
long-- So if I cut it once,

00:01:46.040 --> 00:01:48.050
now this is a whole unit.

00:01:48.050 --> 00:01:52.140
Now if I cut a whole,
now if I cut it twice,

00:01:52.140 --> 00:01:55.200
so now if I cut 4 times, so
I've cut my original block

00:01:55.200 --> 00:01:56.160
into 4 times.

00:01:56.160 --> 00:02:00.610
So now this right over
here is half a unit.

00:02:00.610 --> 00:02:02.705
And actually, if I want
to convert it into eighths

00:02:02.705 --> 00:02:04.240
I need to cut it 16 times.

00:02:04.240 --> 00:02:06.150
So let me do that.

00:02:06.150 --> 00:02:07.470
15, 16.

00:02:07.470 --> 00:02:11.820
So I cut 2 units
into 16 equal pieces.

00:02:11.820 --> 00:02:16.485
Or I guess this is 1/16 of 2, so
that is going to be an eighth.

00:02:16.485 --> 00:02:17.985
And now I just have
to multiply this

00:02:17.985 --> 00:02:19.318
the appropriate number of times.

00:02:19.318 --> 00:02:20.850
This right over here is 5/8.

00:02:20.850 --> 00:02:22.090
This is 1/8.

00:02:22.090 --> 00:02:26.910
So let me copy
this, 1, 2, 3, 4, 5.

00:02:26.910 --> 00:02:28.600
So that is 5/8.

00:02:28.600 --> 00:02:31.250
And then here, I
need to do 1 and 3/4.

00:02:31.250 --> 00:02:35.350
Remember, 1 and 3/4 is
the same thing as 7/4.

00:02:35.350 --> 00:02:38.190
It's 4/4 plus 3/4, which is 7/4.

00:02:38.190 --> 00:02:41.300
7/4 is the same thing as 14/8.

00:02:41.300 --> 00:02:42.960
So this right over
here is 1/8, I just

00:02:42.960 --> 00:02:44.370
need to make 14 of these.

00:02:44.370 --> 00:02:47.372
2, 3, 4, 5, 6, 7, 8.

00:02:47.372 --> 00:02:49.080
Actually, let me show
you what I'm doing.

00:02:49.080 --> 00:02:52.104
9, 10, 11, 12, 13, 14.

00:02:52.104 --> 00:02:52.770
And there we go.

00:02:52.770 --> 00:02:56.070
That's exactly the same
length as goal block 2.

00:02:56.070 --> 00:02:57.910
This is 14/8.

00:02:57.910 --> 00:02:59.930
14/8 is the exact
same length as this.

00:02:59.930 --> 00:03:01.970
14/8 is the same
thing as 7/4, which

00:03:01.970 --> 00:03:04.150
is the same thing as 1 and 3/4.

00:03:04.150 --> 00:03:09.370
This is 5/8, which is
the same thing as 5/8.

00:03:09.370 --> 00:03:10.680
So let's check our answer.

00:03:10.680 --> 00:03:12.665
And we got it right.


Writing algebraic expressions example 2

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:03.980
First, consider the
expression for take

00:00:03.980 --> 00:00:08.515
the quantity of negative
4 times x and add 9.

00:00:08.515 --> 00:00:09.890
So let's see,
we're going to take

00:00:09.890 --> 00:00:12.980
the quantity of
negative 4 times x.

00:00:12.980 --> 00:00:16.764
Well, negative 4 times x is
just going to be negative 4x.

00:00:16.764 --> 00:00:18.180
And then they say--
so we're going

00:00:18.180 --> 00:00:20.555
to take that quantity, that's
this thing right over here,

00:00:20.555 --> 00:00:22.400
and then they say add 9.

00:00:22.400 --> 00:00:26.120
So let's add 9 to it, so
that's this first sentence.

00:00:26.120 --> 00:00:28.350
Take the quantity of
negative 4 times x and add 9.

00:00:28.350 --> 00:00:30.630
That's this expression
right over here.

00:00:30.630 --> 00:00:32.229
Now select the
answer that matches

00:00:32.229 --> 00:00:36.480
the following-- the sum of 4
and the product of negative 9

00:00:36.480 --> 00:00:37.817
and that expression.

00:00:37.817 --> 00:00:38.900
So let's think about this.

00:00:38.900 --> 00:00:43.040
The sum of 4, so
we're going to take 4

00:00:43.040 --> 00:00:48.360
and we're going to add it to the
product-- we're going to add it

00:00:48.360 --> 00:00:49.820
to-- I'm running out of colors.

00:00:49.820 --> 00:00:51.536
Maybe I will do this
in this-- let' see.

00:00:51.536 --> 00:00:53.970
I've already used blue and
I've already used orange.

00:00:53.970 --> 00:00:55.480
I'll use green.

00:00:55.480 --> 00:00:58.200
So we're going to take
that 4, the sum of 4

00:00:58.200 --> 00:01:03.020
and the product of negative
9 and that expression.

00:01:03.020 --> 00:01:05.050
So we're going to add 4
to the product-- we're

00:01:05.050 --> 00:01:10.450
going to multiply negative 9
times this expression right up

00:01:10.450 --> 00:01:11.250
here.

00:01:11.250 --> 00:01:14.490
So it's negative 9
times negative 4x,

00:01:14.490 --> 00:01:19.340
negative 4x plus 9.

00:01:19.340 --> 00:01:21.230
Now let's see if any
of these actually

00:01:21.230 --> 00:01:23.980
match what I just
wrote right over here.

00:01:23.980 --> 00:01:25.879
So let's see, this one is 4.

00:01:25.879 --> 00:01:27.670
Here we're taking a 4
and we're multiplying

00:01:27.670 --> 00:01:30.300
it times negative
4x minus 9, and then

00:01:30.300 --> 00:01:32.600
we're subtracting that, so
this seems very different,

00:01:32.600 --> 00:01:35.410
so I'll scratch that one out.

00:01:35.410 --> 00:01:36.970
Let's see.

00:01:36.970 --> 00:01:39.910
This has positive 9 times
negative 4x plus 9 plus 4,

00:01:39.910 --> 00:01:41.910
so this is very close,
but this says positive 9,

00:01:41.910 --> 00:01:45.050
this says negative 9, so
I'll cross out one out.

00:01:45.050 --> 00:01:47.510
This one has 4 times 9x plus 4.

00:01:47.510 --> 00:01:49.730
Well, no, we have
negative 4x plus 9,

00:01:49.730 --> 00:01:52.280
this is different
in multiple ways,

00:01:52.280 --> 00:01:55.150
and they're multiplying it
by 4 instead of negative 9.

00:01:55.150 --> 00:02:00.570
This one, let's see, we have
a negative 9 being multiplied

00:02:00.570 --> 00:02:07.550
by negative 4x plus 9,
and then we are adding 4.

00:02:07.550 --> 00:02:09.560
So all you have to do to
go from this expression

00:02:09.560 --> 00:02:14.050
to this expression, instead
of adding the 4 there,

00:02:14.050 --> 00:02:16.630
you just add the 4
here, and then these two

00:02:16.630 --> 00:02:18.740
are identical expressions.

00:02:18.740 --> 00:02:22.990
So we will go with this
one right over here.


Fractional exponents with numerators other than 1

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:03.950
We've already seen how to
think about something like 64

00:00:03.950 --> 00:00:05.340
to the 1/3 power.

00:00:05.340 --> 00:00:07.630
We saw that this is
the exact same thing

00:00:07.630 --> 00:00:11.630
as taking the cube root of 64.

00:00:11.630 --> 00:00:14.090
And because we know
that 4 times 4 times

00:00:14.090 --> 00:00:17.957
4, or 4 to the third
power, is equal to 64,

00:00:17.957 --> 00:00:19.665
if we're looking for
the cube root of 64,

00:00:19.665 --> 00:00:22.830
we're looking for a number that
that number times that number

00:00:22.830 --> 00:00:24.935
times that same number is
going to be equal to 64.

00:00:24.935 --> 00:00:28.420
Well, we know that number is 4,
so this thing right over here

00:00:28.420 --> 00:00:30.140
is going to be 4.

00:00:30.140 --> 00:00:32.220
Now we're going to
think of slightly more

00:00:32.220 --> 00:00:34.710
complex fractional exponents.

00:00:34.710 --> 00:00:37.129
The one we see here has
a 1 in the numerator.

00:00:37.129 --> 00:00:38.920
Now we're going to see
something different.

00:00:38.920 --> 00:00:46.470
So what I want to do is think
about what 64 to the 2/3 power

00:00:46.470 --> 00:00:47.280
is.

00:00:47.280 --> 00:00:49.520
And here I'm going to use
a property of exponents

00:00:49.520 --> 00:00:51.600
that we'll study more later on.

00:00:51.600 --> 00:00:53.580
But this property
of exponents is

00:00:53.580 --> 00:00:55.610
the idea that-- let's say
with a simple number--

00:00:55.610 --> 00:00:58.230
if I raise something
to the third power

00:00:58.230 --> 00:01:01.160
and then I were to raise that
to, say, the fourth power,

00:01:01.160 --> 00:01:04.720
this is going to be the same
thing as raising it to the 2

00:01:04.720 --> 00:01:09.650
to the 3 times 4 power, or 2 to
the 12th power, which you could

00:01:09.650 --> 00:01:14.410
also write as raising it to
the fourth power and then

00:01:14.410 --> 00:01:15.160
the third power.

00:01:15.160 --> 00:01:17.410
All this is saying is, if I
raise something to a power

00:01:17.410 --> 00:01:19.390
and then raise that
whole thing to a power,

00:01:19.390 --> 00:01:22.000
it's the same thing as
multiplying the two exponents.

00:01:22.000 --> 00:01:23.950
This is the same thing
as 2 to the 12th.

00:01:23.950 --> 00:01:25.740
So we could use
that property here

00:01:25.740 --> 00:01:29.777
to say, well, 2/3 is the
same thing as 1/3 times 2.

00:01:29.777 --> 00:01:31.360
So we could go in
the other direction.

00:01:31.360 --> 00:01:32.735
We could say, hey
look, well this

00:01:32.735 --> 00:01:36.070
is going to be the same
thing as 64 to the 1/3 power

00:01:36.070 --> 00:01:38.154
and then that thing squared.

00:01:38.154 --> 00:01:39.820
Notice, I'm raising
something to a power

00:01:39.820 --> 00:01:41.250
and then raising
that to a power.

00:01:41.250 --> 00:01:42.875
If I were to multiply
these two things,

00:01:42.875 --> 00:01:46.310
I would get 64 to the 2/3 power.

00:01:46.310 --> 00:01:47.850
Now, why did I do this?

00:01:47.850 --> 00:01:50.554
Well, we already know what
64 to the 1/3 power is.

00:01:50.554 --> 00:01:51.470
We just calculated it.

00:01:51.470 --> 00:01:53.190
That's equal to 4.

00:01:53.190 --> 00:01:55.695
So we could say that--
and I'll write it

00:01:55.695 --> 00:01:57.870
in that same yellow
color-- this is

00:01:57.870 --> 00:02:05.425
equal to 4 squared,
which is equal to 16.

00:02:08.210 --> 00:02:11.234
So 64 to the 2/3
power is equal to 16.

00:02:11.234 --> 00:02:12.650
The way I think
of it, let me find

00:02:12.650 --> 00:02:14.720
the cube root of 64, which is 4.

00:02:14.720 --> 00:02:16.190
And then let me square it.

00:02:16.190 --> 00:02:18.940
And that is going
to get me to 16.

00:02:18.940 --> 00:02:21.250
Now I'll give you in
even hairier problem.

00:02:21.250 --> 00:02:23.510
And I encourage you to
try this one on your own

00:02:23.510 --> 00:02:25.860
before I work through it.

00:02:25.860 --> 00:02:27.440
So we're going to
work with 8/27.

00:02:31.940 --> 00:02:36.790
And we're going to raise
this thing to the--

00:02:36.790 --> 00:02:42.450
and I'll try to color code
it-- negative 2 over 3 power,

00:02:42.450 --> 00:02:44.810
to the negative 2/3 power.

00:02:44.810 --> 00:02:48.814
I encourage you to pause
and try this on your own.

00:02:48.814 --> 00:02:50.230
Well the first
thing I do whenever

00:02:50.230 --> 00:02:52.120
I see a negative
exponent is to say, well,

00:02:52.120 --> 00:02:53.720
how can I get rid of
that negative exponent?

00:02:53.720 --> 00:02:56.219
And I just remind myself, well,
the negative exponent really

00:02:56.219 --> 00:02:57.840
just says, take the
reciprocal of this

00:02:57.840 --> 00:02:58.881
to the positive exponent.

00:03:05.580 --> 00:03:07.210
I'm using a different color.

00:03:07.210 --> 00:03:09.240
I'm going to use that
light mauve color.

00:03:09.240 --> 00:03:10.950
So this is going to
be equal to 27/8.

00:03:13.730 --> 00:03:16.660
I just took the reciprocal
of this right over here.

00:03:16.660 --> 00:03:20.980
It's equal to 27/8 to
the positive 2/3 power.

00:03:23.550 --> 00:03:26.560
So notice, all I did, I
got rid of the exponent

00:03:26.560 --> 00:03:29.010
and took the reciprocal of
the base right over here.

00:03:29.010 --> 00:03:30.630
8/27 is the base.

00:03:30.630 --> 00:03:33.550
Negative 2/3 is the exponent.

00:03:33.550 --> 00:03:35.900
Now, how can we handle this?

00:03:35.900 --> 00:03:38.160
Well, we've already
seen that if I

00:03:38.160 --> 00:03:40.890
have a numerator to some
power over a denominator

00:03:40.890 --> 00:03:43.770
to some power-- and this is
another very powerful exponent

00:03:43.770 --> 00:03:49.630
property-- this is going to be
the exact same thing as raising

00:03:49.630 --> 00:04:00.300
27 to the 2/3 power-- to
the 2 over 3 power-- over 8

00:04:00.300 --> 00:04:01.320
to the 2/3 power.

00:04:04.820 --> 00:04:07.300
This is another very
powerful exponent property.

00:04:07.300 --> 00:04:09.300
Notice, if I have something
divided by something

00:04:09.300 --> 00:04:10.980
and I'm raising the
whole thing to a power,

00:04:10.980 --> 00:04:12.521
I can essentially
raise the numerator

00:04:12.521 --> 00:04:14.340
to that power over
the denominator raised

00:04:14.340 --> 00:04:15.430
to that power.

00:04:15.430 --> 00:04:17.714
Now, let's think
about what this is.

00:04:17.714 --> 00:04:19.130
Well just like we
saw before, this

00:04:19.130 --> 00:04:20.930
is going to be the same thing.

00:04:20.930 --> 00:04:29.060
This is going to be the same
thing as 27 to the 1/3 power

00:04:29.060 --> 00:04:34.432
and then that squared
because 1/3 times 2 is 2/3.

00:04:34.432 --> 00:04:36.140
So I'm going to raise
27 to the 1/3 power

00:04:36.140 --> 00:04:40.050
and then square
whatever that is.

00:04:40.050 --> 00:04:42.539
All this color
coding is making this

00:04:42.539 --> 00:04:43.830
have to switch a lot of colors.

00:04:43.830 --> 00:04:47.620
This is going to be
over 8 to the 1/3 power.

00:04:51.829 --> 00:04:54.120
And then that's going to be
raised to the second power.

00:04:54.120 --> 00:04:55.995
Same thing we were doing
in the denominator--

00:04:55.995 --> 00:04:58.030
we raise 8 to the 1/3
and then square that.

00:04:58.030 --> 00:04:59.370
So what's this going to be?

00:04:59.370 --> 00:05:05.670
Well, 27 to the 1/3 power
is the cube root of 27.

00:05:05.670 --> 00:05:08.060
It's some number-- that
number times that same number

00:05:08.060 --> 00:05:10.580
times that same number is
going to be equal to 27.

00:05:10.580 --> 00:05:13.010
Well, it might jump out at you
already that 3 to the third

00:05:13.010 --> 00:05:17.350
is equal to 27 or that 27
to the 1/3 is equal to 3.

00:05:17.350 --> 00:05:21.310
So the numerator, we're going
to end up with 3 squared.

00:05:21.310 --> 00:05:23.730
And then in the
denominator, we are

00:05:23.730 --> 00:05:27.050
going to end up with-- well,
what's 8 to the 1/3 power?

00:05:27.050 --> 00:05:29.720
Well, 2 times 2 times 2 is 8.

00:05:29.720 --> 00:05:31.825
So this is 8 to
the 1/3 third is 2.

00:05:34.780 --> 00:05:36.430
Let me do that
same orange color.

00:05:36.430 --> 00:05:41.160
8 to the 1/3 is 2, and then
we're going to square that.

00:05:41.160 --> 00:05:43.320
So this is going to
simplify to 3 squared

00:05:43.320 --> 00:05:47.611
over 2 squared, which is just
going to be equal to 9/4.

00:05:47.611 --> 00:05:49.360
So if you just break
it down step by step,

00:05:49.360 --> 00:05:52.535
it actually is not too daunting.


One-sided limits from graphs

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.180
So if we were to
ask ourselves, what

00:00:03.180 --> 00:00:08.860
is the value of our
function approaching--

00:00:08.860 --> 00:00:12.870
as we approach x equals 2 from
values less than x equals 2.

00:00:12.870 --> 00:00:15.330
So as you imagine, as
we approach x equals 2--

00:00:15.330 --> 00:00:20.860
So x equals 1, x equals 1.5,
x equals 1.9, x equals 1.999,

00:00:20.860 --> 00:00:23.520
x equals 1.99999999.

00:00:23.520 --> 00:00:25.540
What is f of x approaching?

00:00:25.540 --> 00:00:30.390
And we see that f of x
seems to be approaching

00:00:30.390 --> 00:00:33.780
this value right over here.

00:00:33.780 --> 00:00:35.860
It seems to be approaching 5.

00:00:35.860 --> 00:00:38.160
And so the way we
would denote that is

00:00:38.160 --> 00:00:45.510
the limit of f of x,
as x approaches 2--

00:00:45.510 --> 00:00:48.660
and we're going to specify the
direction-- as x approaches 2

00:00:48.660 --> 00:00:50.500
from the negative
direction-- we put

00:00:50.500 --> 00:00:53.462
the negative as a
superscript after the 2

00:00:53.462 --> 00:00:55.420
to denote the direction
that we're approaching.

00:00:55.420 --> 00:00:56.700
This is not a negative 2.

00:00:56.700 --> 00:00:59.510
We're approaching 2 from
the negative direction.

00:00:59.510 --> 00:01:01.960
We're approaching 2
from values less than 2.

00:01:01.960 --> 00:01:05.640
We're getting closer and closer
to 2, but from below-- 1.9,

00:01:05.640 --> 00:01:08.180
1.99, 1.99999 .

00:01:08.180 --> 00:01:10.850
As x gets closer and
closer from those values,

00:01:10.850 --> 00:01:12.590
what is f of x approaching?

00:01:12.590 --> 00:01:18.460
And we see here that
it is approaching 5.

00:01:18.460 --> 00:01:24.490
But what if we were asked the
natural other question-- What

00:01:24.490 --> 00:01:31.320
is the limit of f of
x as x approaches 2

00:01:31.320 --> 00:01:33.590
from values greater than 2?

00:01:33.590 --> 00:01:36.510
So this is a little superscript
positive right over here.

00:01:36.510 --> 00:01:38.484
So now we're going to
approach x equals 2,

00:01:38.484 --> 00:01:40.650
but we're going to approach
it from this direction--

00:01:40.650 --> 00:01:45.550
x equals 3, x equals 2.5, x
equals 2.1, x equals 2.01,

00:01:45.550 --> 00:01:47.194
x equals 2.0001.

00:01:47.194 --> 00:01:49.110
And we're going to get
closer and closer to 2,

00:01:49.110 --> 00:01:52.690
but we're coming from values
that are larger than 2.

00:01:52.690 --> 00:01:55.290
So here, when x equals
3, f of x is here.

00:01:55.290 --> 00:01:59.310
When x equals 2.5,
f of x is here.

00:01:59.310 --> 00:02:04.650
When x equals 2.01, f of x
looks like it's right over here.

00:02:04.650 --> 00:02:09.400
So in this situation, we're
getting closer and closer

00:02:09.400 --> 00:02:11.100
to f of x equaling 1.

00:02:11.100 --> 00:02:12.717
It never does quite equal that.

00:02:12.717 --> 00:02:14.675
It actually then just
has a jump discontinuity.

00:02:17.280 --> 00:02:21.680
This seems to be the limiting
value when we approach when

00:02:21.680 --> 00:02:23.820
we approach 2 from
values greater than 2.

00:02:23.820 --> 00:02:28.470
So this right over
here is equal to 1.

00:02:28.470 --> 00:02:30.660
And so when we think
about limits in general,

00:02:30.660 --> 00:02:33.880
the only way that a limit
at 2 will actually exist

00:02:33.880 --> 00:02:37.410
is if both of these
one-sided limits

00:02:37.410 --> 00:02:39.040
are actually the same thing.

00:02:39.040 --> 00:02:40.430
In this situation, they aren't.

00:02:40.430 --> 00:02:44.200
As we approach 2
from values below 2,

00:02:44.200 --> 00:02:46.400
the function seems
to be approaching 5.

00:02:46.400 --> 00:02:50.130
And as we approach 2
from values above 2,

00:02:50.130 --> 00:02:52.580
the function seems
to be approaching 1.

00:02:52.580 --> 00:02:56.150
So in this case, the limit--
let me write this down--

00:02:56.150 --> 00:03:01.140
the limit of f of
x, as x approaches 2

00:03:01.140 --> 00:03:03.260
from the negative
direction, does not

00:03:03.260 --> 00:03:09.850
equal the limit of f
of x, as x approaches 2

00:03:09.850 --> 00:03:11.280
from the positive direction.

00:03:11.280 --> 00:03:14.570
And since this is the case--
that they're not equal--

00:03:14.570 --> 00:03:16.880
the limit does not exist.

00:03:16.880 --> 00:03:19.650
The limit as x
approaches 2 in general

00:03:19.650 --> 00:03:24.930
of f of x-- so the limit of
f of x, as x approaches 2,

00:03:24.930 --> 00:03:28.310
does not exist.

00:03:28.310 --> 00:03:31.310
In order for it to have
existed, these two things

00:03:31.310 --> 00:03:34.030
would have had to have
been equal to each other.

00:03:34.030 --> 00:03:35.930
For example, if
someone were to say,

00:03:35.930 --> 00:03:44.760
what is the limit of f
of x as x approaches 4?

00:03:44.760 --> 00:03:50.100
Well, then we could think about
the two one-sided limits--

00:03:50.100 --> 00:03:53.340
the one-sided limit from
below and the one-sided limit

00:03:53.340 --> 00:03:54.230
from above.

00:03:54.230 --> 00:03:55.605
So we could say,
well, let's see.

00:03:55.605 --> 00:04:00.780
The limit of f of x, as x
approaches 4 from below-- so

00:04:00.780 --> 00:04:01.610
let me draw that.

00:04:01.610 --> 00:04:04.380
So what we care
about-- x equals 4.

00:04:04.380 --> 00:04:07.280
As x equals 4 from below--

00:04:07.280 --> 00:04:10.660
So when x equals 3,
we're here where f of 3

00:04:10.660 --> 00:04:15.850
is negative 2. f of 3.5
seems to be right over here.

00:04:15.850 --> 00:04:20.170
f of 3.9 seems to be right
over here. f of 3.999--

00:04:20.170 --> 00:04:24.360
we're getting closer and
closer to our function equaling

00:04:24.360 --> 00:04:26.050
negative 5.

00:04:26.050 --> 00:04:29.760
So the limit as we
approach 4 from below--

00:04:29.760 --> 00:04:31.640
this one-sided
limit from the left,

00:04:31.640 --> 00:04:36.320
we could say-- this is going
to be equal to negative 5.

00:04:36.320 --> 00:04:38.740
And if we were to ask
ourselves the limit of f

00:04:38.740 --> 00:04:42.470
of x, as x approaches
4 from the right,

00:04:42.470 --> 00:04:48.330
from values larger than
4, well, same exercise.

00:04:48.330 --> 00:04:51.060
f of 5 gets us here.

00:04:51.060 --> 00:04:53.600
f of 4.5 seems
right around here.

00:04:53.600 --> 00:04:57.950
f of 4.1 seems right
about here. f of 4.01

00:04:57.950 --> 00:04:59.000
seems right around here.

00:04:59.000 --> 00:05:00.499
And even f of 4 is
actually defined,

00:05:00.499 --> 00:05:02.370
but we're getting
closer and closer to it.

00:05:02.370 --> 00:05:05.670
And we see, once again,
we are approaching 5.

00:05:05.670 --> 00:05:08.590
Even if f of 4 was not
defined on either side,

00:05:08.590 --> 00:05:12.480
we would be
approaching negative 5.

00:05:12.480 --> 00:05:14.800
So this is also
approaching negative 5.

00:05:14.800 --> 00:05:17.720
And since the limit
from the left-hand side

00:05:17.720 --> 00:05:20.130
is equal to the limit
from the right-hand side,

00:05:20.130 --> 00:05:22.880
we can say-- so these
two things are equal.

00:05:22.880 --> 00:05:24.870
And because these
two things are equal,

00:05:24.870 --> 00:05:27.830
we know that the limit of
f of x, as x approaches 4,

00:05:27.830 --> 00:05:30.300
is equal to 5.

00:05:30.300 --> 00:05:33.690
Let's look at a
few more examples.

00:05:33.690 --> 00:05:41.920
So let's ask ourselves
the limit of f of x-- now,

00:05:41.920 --> 00:05:47.600
this is our new f of x depicted
here-- as x approaches 8.

00:05:47.600 --> 00:05:51.360
And let's approach
8 from the left.

00:05:51.360 --> 00:05:54.583
As x approaches 8 from
values less than 8.

00:05:54.583 --> 00:05:56.083
So what's this going
to be equal to?

00:05:56.083 --> 00:05:58.666
And I encourage you to pause the
video to try to figure it out

00:05:58.666 --> 00:05:59.220
yourself.

00:05:59.220 --> 00:06:01.210
So x is getting closer
and closer to 8.

00:06:01.210 --> 00:06:04.050
So if x is 7, f of 7 is here.

00:06:04.050 --> 00:06:07.840
If x is 7.5, f of 7.5 is here.

00:06:07.840 --> 00:06:09.920
So it looks like
our value of f of x

00:06:09.920 --> 00:06:14.740
is getting closer and
closer and closer to 3.

00:06:14.740 --> 00:06:17.430
So it looks like the limit
of f of x, as x approaches 8

00:06:17.430 --> 00:06:21.290
from the negative
side, is equal to 3.

00:06:21.290 --> 00:06:22.910
What about from
the positive side?

00:06:22.910 --> 00:06:26.500
What about the
limit of f of x as x

00:06:26.500 --> 00:06:28.790
approaches 8 from the
positive direction

00:06:28.790 --> 00:06:30.220
or from the right side?

00:06:30.220 --> 00:06:34.130
Well, here we see as x
is 9, this is our f of x.

00:06:34.130 --> 00:06:37.950
As x is 8.5, this
is our f of 8.5.

00:06:37.950 --> 00:06:43.960
It seems like we're
approaching f of x equaling 1.

00:06:43.960 --> 00:06:46.620
So notice, these two
limits are different.

00:06:46.620 --> 00:06:51.400
So the non-one-sided limit,
or the two-sided limit,

00:06:51.400 --> 00:06:55.677
does not exist at f of
x or as we approach 8.

00:06:55.677 --> 00:06:56.760
So let me write that down.

00:06:56.760 --> 00:07:01.330
The limit of f of x,
as x approaches 8--

00:07:01.330 --> 00:07:04.370
because these two things
are not the same value--

00:07:04.370 --> 00:07:05.875
this does not exist.

00:07:10.540 --> 00:07:11.870
Let's do one more example.

00:07:11.870 --> 00:07:14.120
And here they're actually
asking us a question.

00:07:14.120 --> 00:07:16.290
The function f is graphed below.

00:07:16.290 --> 00:07:20.420
What appears to be the value of
the one-sided limit, the limit

00:07:20.420 --> 00:07:26.900
of f of x-- this is f of x--
as x approaches negative 2

00:07:26.900 --> 00:07:28.120
from the negative direction?

00:07:28.120 --> 00:07:30.500
So this is the negative 2
from the negative direction.

00:07:30.500 --> 00:07:32.900
So we care what happens as
x approaches negative 2.

00:07:32.900 --> 00:07:35.542
We see f of x is actually
undefined right over there.

00:07:35.542 --> 00:07:37.250
But let's see what
happens as we approach

00:07:37.250 --> 00:07:39.166
from the negative
direction, or as we approach

00:07:39.166 --> 00:07:40.870
from values less
than negative 2,

00:07:40.870 --> 00:07:43.080
or as we approach from the left.

00:07:43.080 --> 00:07:46.450
As we approach from the
left, f of negative 4

00:07:46.450 --> 00:07:47.560
is right over here.

00:07:47.560 --> 00:07:50.330
So this is f of negative 4.

00:07:50.330 --> 00:07:52.900
f of negative 3 is
right over here.

00:07:55.880 --> 00:07:59.930
f of negative 2.5 seems
to be right over here.

00:07:59.930 --> 00:08:01.770
We seem to be getting
closer and closer

00:08:01.770 --> 00:08:04.840
to f of x being equal
to 4, at least visually.

00:08:04.840 --> 00:08:06.550
So I would say that
it looks-- at least,

00:08:06.550 --> 00:08:10.470
graphically-- the limit of
f of x, as x approaches 2

00:08:10.470 --> 00:08:14.670
from the negative
direction, is equal to 4.

00:08:14.670 --> 00:08:17.210
Now, if we also asked
ourselves the limit

00:08:17.210 --> 00:08:20.680
of f of x, as x
approaches negative 2

00:08:20.680 --> 00:08:24.267
from the positive direction,
we would get a similar result.

00:08:24.267 --> 00:08:25.850
Now, we're going to
approach from when

00:08:25.850 --> 00:08:29.470
x is 0, f of x seems
to be right over here.

00:08:29.470 --> 00:08:32.330
When x is 1, f of x
is right over here.

00:08:36.740 --> 00:08:39.350
When x is negative
1, f of x is there.

00:08:39.350 --> 00:08:43.809
When x is negative 1.9, f of
x seems to be right over here.

00:08:43.809 --> 00:08:47.400
So once again, we seem to be
getting closer and closer to 4.

00:08:47.400 --> 00:08:50.240
Because the left-handed limit
and the right-handed limit

00:08:50.240 --> 00:08:51.040
are the same value.

00:08:51.040 --> 00:08:54.180
Because both one-sided limits
are approaching the same thing,

00:08:54.180 --> 00:09:01.240
we can say that the limit
of f of x, as x approaches

00:09:01.240 --> 00:09:03.375
negative 2-- and this
is from both directions.

00:09:03.375 --> 00:09:06.090
Since from both directions, we
get the same limiting value,

00:09:06.090 --> 00:09:07.770
we can say that the
limit exists there.

00:09:07.770 --> 00:09:10.710
And it is equal to 4.


Interpreting slope of a curve exercise

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:03.280
A function f of x
is plotted below.

00:00:03.280 --> 00:00:05.560
Highlight an interval
where f prime

00:00:05.560 --> 00:00:08.480
of x with the first derivative
of f with respect to x

00:00:08.480 --> 00:00:10.250
is greater than 0.

00:00:10.250 --> 00:00:13.390
So if our derivative, f prime
of x, is greater than 0,

00:00:13.390 --> 00:00:16.230
that means that the slope of
the tangent line is positive.

00:00:16.230 --> 00:00:17.660
That means that
the function must

00:00:17.660 --> 00:00:20.610
be increasing
whenever this is true.

00:00:20.610 --> 00:00:22.320
So where is the
function increasing?

00:00:22.320 --> 00:00:25.760
So over here we see
that as x increases,

00:00:25.760 --> 00:00:28.450
our function is going to
smaller and smaller values.

00:00:28.450 --> 00:00:31.580
So our function is
decreasing this entire place.

00:00:31.580 --> 00:00:34.670
Then it's decreasing at
slower and slower rates

00:00:34.670 --> 00:00:36.370
all the way to the
point right over here

00:00:36.370 --> 00:00:39.919
where the slope of the
tangent line is flat.

00:00:39.919 --> 00:00:41.585
And then the function
starts increasing.

00:00:41.585 --> 00:00:44.200
And it starts increasing at
faster and faster and faster

00:00:44.200 --> 00:00:44.980
rates.

00:00:44.980 --> 00:00:48.417
So we could put this blue
area anywhere from here

00:00:48.417 --> 00:00:50.500
to anywhere in this region
here where the function

00:00:50.500 --> 00:00:51.680
itself is increasing.

00:00:51.680 --> 00:00:53.870
Notice, the function
could even be negative,

00:00:53.870 --> 00:00:55.190
but the function is increasing.

00:00:55.190 --> 00:00:56.884
So even this would
be a valid place.

00:00:56.884 --> 00:00:58.050
That would be a valid place.

00:00:58.050 --> 00:01:01.960
Anywhere where the function's
rate of-- when x increases,

00:01:01.960 --> 00:01:04.280
the function is
actually getting larger.

00:01:04.280 --> 00:01:05.560
So let's check our answer.

00:01:05.560 --> 00:01:08.390
And let's do a
couple more of these.

00:01:08.390 --> 00:01:10.410
A function f of x
is plotted below.

00:01:10.410 --> 00:01:13.260
Highlight an interval where
f of x is greater than 0.

00:01:13.260 --> 00:01:16.304
So f of x should be sitting in
the first or second quadrant

00:01:16.304 --> 00:01:16.970
right over here.

00:01:16.970 --> 00:01:19.410
It needs to be greater
than 0, and f prime

00:01:19.410 --> 00:01:21.550
of x is less than 0.

00:01:21.550 --> 00:01:24.140
So f prime of x
less than 0 means

00:01:24.140 --> 00:01:26.270
that the function is decreasing.

00:01:26.270 --> 00:01:28.290
The slope of the tangent
line is negative.

00:01:28.290 --> 00:01:29.290
So let's think about it.

00:01:29.290 --> 00:01:32.010
The two areas where f
of x is greater than 0

00:01:32.010 --> 00:01:34.820
are this interval
right over here

00:01:34.820 --> 00:01:37.840
and this interval
right over here.

00:01:37.840 --> 00:01:40.990
But we also care about
the function decreasing.

00:01:40.990 --> 00:01:43.780
This won't be valid, because
the function is increasing here.

00:01:43.780 --> 00:01:45.571
It's increasing at
slower and slower rates,

00:01:45.571 --> 00:01:46.480
but it is increasing.

00:01:46.480 --> 00:01:49.420
The function is going
up as x goes up.

00:01:49.420 --> 00:01:54.810
If we go over here, the function
is decreasing as x increases.

00:01:54.810 --> 00:01:56.750
So this seems to
meet our constraints.

00:01:56.750 --> 00:01:59.660
This area right over here
won't because the function

00:01:59.660 --> 00:02:01.680
is positive, but
it's also increasing.

00:02:01.680 --> 00:02:04.000
So here the derivative
is positive.

00:02:04.000 --> 00:02:05.990
Here the derivative is positive.

00:02:05.990 --> 00:02:08.199
Here the derivative is negative.

00:02:08.199 --> 00:02:10.180
So the function
itself is positive,

00:02:10.180 --> 00:02:13.070
but the slope of the
tangent line is negative.

00:02:13.070 --> 00:02:16.820
It's decreasing as x increases.

00:02:16.820 --> 00:02:19.600
Let's do one more.

00:02:19.600 --> 00:02:21.050
The function f is plotted below.

00:02:21.050 --> 00:02:23.600
Highlight an interval where
f of x is greater than 0

00:02:23.600 --> 00:02:24.970
and f prime of x is less than 0.

00:02:24.970 --> 00:02:27.820
So the same exact idea--
where are we positive

00:02:27.820 --> 00:02:29.910
but our function is
actually decreasing?

00:02:29.910 --> 00:02:32.320
So it's positive here,
but we can't pick that.

00:02:32.320 --> 00:02:34.660
It's positive here,
but we can't pick that.

00:02:34.660 --> 00:02:38.210
The function is decreasing here,
but the function isn't positive

00:02:38.210 --> 00:02:38.710
there.

00:02:38.710 --> 00:02:41.210
The derivative is negative, but
the function isn't positive.

00:02:41.210 --> 00:02:43.630
So that wouldn't make it.

00:02:43.630 --> 00:02:48.240
So this is the only region
that we can throw it in.

00:02:48.240 --> 00:02:50.640
And we got it right again.


Recognizing concavity exercise

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:03.330
A function f of x
is plotted below.

00:00:03.330 --> 00:00:06.680
Highlight an interval
where f prime of x, or we

00:00:06.680 --> 00:00:08.870
could say the first
derivative of x,

00:00:08.870 --> 00:00:11.260
for the first derivative
of f with respect

00:00:11.260 --> 00:00:14.750
to x is greater than
0 and f double prime

00:00:14.750 --> 00:00:17.710
of x, or the second derivative
of f with respect to x,

00:00:17.710 --> 00:00:18.770
is less than 0.

00:00:18.770 --> 00:00:20.560
So let's think about
what they're saying.

00:00:20.560 --> 00:00:22.851
So we're looking for a place
where the first derivative

00:00:22.851 --> 00:00:24.320
is greater than 0.

00:00:24.320 --> 00:00:27.530
That means that the slope of
the tangent line is positive.

00:00:27.530 --> 00:00:31.100
That means that the function is
increasing over that interval.

00:00:31.100 --> 00:00:32.670
So if we just think
about it here,

00:00:32.670 --> 00:00:34.310
over this whole region
right over here,

00:00:34.310 --> 00:00:36.690
the function is
clearly decreasing.

00:00:36.690 --> 00:00:40.130
Then the slope becomes
0 right over here.

00:00:40.130 --> 00:00:42.430
And then the function
starts increasing again,

00:00:42.430 --> 00:00:45.020
all the way until this
point right over here.

00:00:45.020 --> 00:00:46.190
It hits 0.

00:00:46.190 --> 00:00:48.850
And then it goes, and the
function starts decreasing.

00:00:48.850 --> 00:00:51.405
Just this first constraint
right over here tells us

00:00:51.405 --> 00:00:55.050
it's going to be something in
this interval right over there.

00:00:55.050 --> 00:00:57.200
And then they say where
the second derivative

00:00:57.200 --> 00:00:59.570
is less than 0.

00:00:59.570 --> 00:01:03.840
So this means that
the slope itself,

00:01:03.840 --> 00:01:05.290
whether it's
positive or negative,

00:01:05.290 --> 00:01:07.780
that it's actually decreasing.

00:01:07.780 --> 00:01:10.850
We are going to be concave
downwards right over here.

00:01:10.850 --> 00:01:13.120
The slope itself--
it could be positive.

00:01:13.120 --> 00:01:15.979
But it will be becoming less
and less and less positive.

00:01:15.979 --> 00:01:18.520
And so we're looking for a place
where the slope is positive,

00:01:18.520 --> 00:01:21.890
but it's becoming less and
less and less positive.

00:01:21.890 --> 00:01:24.720
If you look over here,
the slope is positive.

00:01:24.720 --> 00:01:25.980
But the slope is increasing.

00:01:25.980 --> 00:01:28.551
It's getting steeper and
steeper and steeper as we go.

00:01:28.551 --> 00:01:30.050
And then, all of a
sudden, it starts

00:01:30.050 --> 00:01:32.420
getting less steep, less
steep, less steep, less steep

00:01:32.420 --> 00:01:35.400
all the way to when the
slope gets back to 0.

00:01:35.400 --> 00:01:37.170
So if we want to
select an interval,

00:01:37.170 --> 00:01:39.470
it would be this
interval right over here.

00:01:39.470 --> 00:01:40.950
Our slope is positive.

00:01:40.950 --> 00:01:43.150
Our function is
clearly increasing,

00:01:43.150 --> 00:01:46.130
but it is increasing at
a lower and lower rate.

00:01:46.130 --> 00:01:48.560
So I will select that
right over there.

00:01:48.560 --> 00:01:51.270
Let's do one more example.

00:01:51.270 --> 00:01:52.780
A function f of x
is plotted below.

00:01:52.780 --> 00:01:56.630
Highlight an interval where f
prime of x is greater than 0.

00:01:56.630 --> 00:01:59.640
So the same thing where
our function is increasing,

00:01:59.640 --> 00:02:02.830
but it's increasing at a
slower and slower rate.

00:02:02.830 --> 00:02:06.720
So our function is increasing
in this whole region

00:02:06.720 --> 00:02:09.360
right over here, and we
see it's really steep here,

00:02:09.360 --> 00:02:11.280
that it's getting less
steep and less steep.

00:02:11.280 --> 00:02:12.696
And it's getting
closer and closer

00:02:12.696 --> 00:02:14.400
to 0, the slope of
the tangent line

00:02:14.400 --> 00:02:17.090
or the rate of increase
of the function.

00:02:17.090 --> 00:02:22.394
So I would pick anything right
around this region right here.


Vertical, adjacent and linearly paired angles

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:05.160
We're asked to name an
angle adjacent to angle BGD.

00:00:05.160 --> 00:00:07.850
So angle BGD, let's see
if we can pick it out.

00:00:07.850 --> 00:00:13.380
So here is B, here is G, and
here is D, right over here.

00:00:13.380 --> 00:00:20.950
So angle BGD is this entire
angle right over here.

00:00:20.950 --> 00:00:22.845
So when we talk about
adjacent angles,

00:00:22.845 --> 00:00:24.220
we're talking
about an angle that

00:00:24.220 --> 00:00:26.680
has one of its rays in common.

00:00:26.680 --> 00:00:32.890
So for example, angle AGB has
one of the rays in common,

00:00:32.890 --> 00:00:35.750
it has GB in common
with angle BGD.

00:00:35.750 --> 00:00:42.540
So we could say angle AGB,
which could obviously also

00:00:42.540 --> 00:00:47.950
be called angle
BGA, BGA and AGB are

00:00:47.950 --> 00:00:50.800
both this angle right over here.

00:00:50.800 --> 00:00:54.550
You could also go
with angle FGB,

00:00:54.550 --> 00:00:56.590
because that also
has GB in common.

00:00:56.590 --> 00:01:02.450
So you go angle FGB, which could
also be written as angle BGF.

00:01:02.450 --> 00:01:08.760
Or you could go over here, angle
EGD shares ray GD in common.

00:01:08.760 --> 00:01:12.251
So you could do this angle
right over here, angle EGD.

00:01:14.890 --> 00:01:17.546
Or you could go all the
way out here, angle FGD.

00:01:20.860 --> 00:01:24.940
These last two sharing
ray GD in common.

00:01:24.940 --> 00:01:29.070
So any one of these responses
would satisfy the question

00:01:29.070 --> 00:01:31.840
of just naming an
angle, just naming one.

00:01:31.840 --> 00:01:33.190
Let's do this next one.

00:01:33.190 --> 00:01:35.823
Name an angle
vertical to angle EGA.

00:01:39.630 --> 00:01:43.790
So this is this angle
right over here.

00:01:43.790 --> 00:01:45.850
And the way you think
about vertical angles

00:01:45.850 --> 00:01:48.390
is, imagine two lines crossing.

00:01:48.390 --> 00:01:52.000
So imagine two lines
crossing, just like this.

00:01:52.000 --> 00:01:55.360
And they could
literally be lines,

00:01:55.360 --> 00:01:57.670
and they're
intersecting at a point.

00:01:57.670 --> 00:02:00.940
This is forming
four angles, or you

00:02:00.940 --> 00:02:04.130
could imagine it's forming
two sets of vertical angles.

00:02:04.130 --> 00:02:06.700
So if this is the angle
that you care about,

00:02:06.700 --> 00:02:09.100
it's a vertical
angle, it's the one

00:02:09.100 --> 00:02:10.870
on the opposite side
of the intersection.

00:02:10.870 --> 00:02:13.990
It's one of these angles
that it is not adjacent to.

00:02:13.990 --> 00:02:16.850
So it would be this
angle right over here.

00:02:16.850 --> 00:02:22.371
So going back to the question,
a vertical angle to angle EGA,

00:02:22.371 --> 00:02:23.870
well if you imagine
the intersection

00:02:23.870 --> 00:02:29.350
of line EB and line DA,
then the non-adjacent angle

00:02:29.350 --> 00:02:33.150
formed to angle
EGA is angle DGB.

00:02:33.150 --> 00:02:35.400
Actually, what we already
highlighted in magenta right

00:02:35.400 --> 00:02:36.250
over here.

00:02:36.250 --> 00:02:38.490
So this is angle DGB.

00:02:41.190 --> 00:02:44.090
Which could also be
called angle BGD.

00:02:44.090 --> 00:02:46.090
These are obviously both
referring to this angle

00:02:46.090 --> 00:02:47.610
up here.

00:02:47.610 --> 00:02:51.875
Name an angle that forms a
linear pair with the angle DFG.

00:02:51.875 --> 00:02:53.250
So we'll put this
in a new color.

00:02:53.250 --> 00:02:54.143
Angle DFG.

00:02:58.938 --> 00:03:01.500
Sorry, DGF, all of these
should have G in the middle.

00:03:01.500 --> 00:03:03.440
DGF.

00:03:03.440 --> 00:03:06.070
So linear pair with
angle DGF, so that's

00:03:06.070 --> 00:03:07.660
this angle right over here.

00:03:07.660 --> 00:03:10.640
So an angle that forms a linear
pair will be an angle that

00:03:10.640 --> 00:03:15.170
is adjacent, where the two outer
rays combined will form a line.

00:03:15.170 --> 00:03:18.630
So for example, if you
combine angle DGF, which

00:03:18.630 --> 00:03:24.110
is this angle, and angle DGC,
then their two outer rays

00:03:24.110 --> 00:03:28.540
form this entire
line right over here.

00:03:28.540 --> 00:03:29.646
So we could say angle DGC.

00:03:34.440 --> 00:03:38.990
Or, if you look at angle DFG,
you could form a line this way.

00:03:38.990 --> 00:03:42.720
If you take angle AGF,
so if you take this one,

00:03:42.720 --> 00:03:44.990
then the outer rays
will form this line.

00:03:44.990 --> 00:03:47.240
So angle AGF would also work.

00:03:47.240 --> 00:03:50.600
Angle AGF.

00:03:50.600 --> 00:03:51.350
Let's do one more.

00:03:51.350 --> 00:03:53.680
Name a vertical
angle to angle FGB.

00:03:57.770 --> 00:04:01.230
So this is FGB right over here.

00:04:01.230 --> 00:04:04.260
You could imagine this angle is
one of the four angles formed

00:04:04.260 --> 00:04:09.570
when CF-- let me highlight
this, that's hard to see.

00:04:12.430 --> 00:04:14.940
This is the last one, so I
can make a mess out of this.

00:04:14.940 --> 00:04:20.936
That angle is formed when CF and
EB intersect with each other.

00:04:20.936 --> 00:04:22.060
And four angles are formed.

00:04:22.060 --> 00:04:25.770
The one question, FGB, these two
angles that are adjacent to it,

00:04:25.770 --> 00:04:27.010
it shares a common ray.

00:04:27.010 --> 00:04:28.520
And then the vertical
angle, the one

00:04:28.520 --> 00:04:30.020
that sits on the opposite side.

00:04:30.020 --> 00:04:34.410
So this angle, this angle right
over here, which is angle EGC.

00:04:34.410 --> 00:04:36.980
Or you could also
call it angle CGE.

00:04:36.980 --> 00:04:39.670
So angle CGE.


Perimeter of a shape

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:02.350
What is the perimeter
of the shape?

00:00:02.350 --> 00:00:05.980
Each square in the grid is
a 1 by 1 centimeter square.

00:00:05.980 --> 00:00:08.160
So all we have to do
is add up the lengths

00:00:08.160 --> 00:00:10.050
of these blue segments
right over here.

00:00:10.050 --> 00:00:11.890
And they put it
on this nice grid.

00:00:11.890 --> 00:00:14.150
And each box here is 1 by 1.

00:00:14.150 --> 00:00:15.820
So let's say we start up here.

00:00:15.820 --> 00:00:18.620
We want to make sure that we
only go to where we started

00:00:18.620 --> 00:00:20.040
and we don't double count.

00:00:20.040 --> 00:00:30.973
So this perimeter is 1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13,

00:00:30.973 --> 00:00:41.780
14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24 centimeters long.

00:00:41.780 --> 00:00:43.474
So it is 24 centimeters.

00:00:43.474 --> 00:00:45.390
The other way we could
have thought about this

00:00:45.390 --> 00:00:47.973
is you could have looked at the
length of each of these sides.

00:00:47.973 --> 00:00:51.620
So this is 1 plus 2
plus 3 plus 1 plus 3

00:00:51.620 --> 00:00:56.530
plus 1 plus-- what is this--
5 plus 2 plus 4 plus 2.

00:00:56.530 --> 00:00:59.560
And you would have also
gotten 24 centimeters.

00:00:59.560 --> 00:01:00.950
Let's do a couple more of these.

00:01:00.950 --> 00:01:02.692
What is the perimeter
of the square?

00:01:02.692 --> 00:01:04.900
So once again, it's the
length of all of the segments

00:01:04.900 --> 00:01:08.350
that define the outside
boundary of the square.

00:01:08.350 --> 00:01:11.480
And by definition, a square--
all of its sides are equal.

00:01:11.480 --> 00:01:15.590
So you have 4 sides that
are all 7 meters long.

00:01:15.590 --> 00:01:17.560
So you could say it's
7 meters plus 7 meters

00:01:17.560 --> 00:01:20.150
plus 7 meters plus
7 meters, or it's

00:01:20.150 --> 00:01:22.725
4 times 7 meters or 28 meters.

00:01:25.240 --> 00:01:26.660
Let's do one more.

00:01:26.660 --> 00:01:29.720
What is the perimeter
of the regular pentagon?

00:01:29.720 --> 00:01:32.950
So it's a regular pentagon,
which means all of its sides

00:01:32.950 --> 00:01:33.969
have the same length.

00:01:33.969 --> 00:01:35.260
And they give us a side of one.

00:01:35.260 --> 00:01:36.350
The length of one is 2.

00:01:36.350 --> 00:01:38.340
So all of the sides
have length 2.

00:01:38.340 --> 00:01:42.860
So it's going to be 2 plus
2 plus 2 plus 2 plus 2,

00:01:42.860 --> 00:01:44.219
or essentially five 2's.

00:01:44.219 --> 00:01:46.010
Or another way of
thinking about it is it's

00:01:46.010 --> 00:01:50.910
5 times 2, which
is going to be 10.

00:01:50.910 --> 00:01:53.130
You have five sides
of this pentagon.

00:01:53.130 --> 00:01:56.080
Each of those sides
are 2 units long.

00:01:56.080 --> 00:02:01.473
So 2 units long times 5 sides
is going to be 10 units.


Dividing completely to get decimal answer example 2

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:05.050
Let's say that we want to
take 14 and divide it by 70.

00:00:05.050 --> 00:00:08.175
So clearly this is going
to be a number less than 1.

00:00:08.175 --> 00:00:11.250
14 is clearly less than
70, but let's try to do it.

00:00:11.250 --> 00:00:13.847
So immediately we say, OK,
70 goes into 1 zero times,

00:00:13.847 --> 00:00:15.180
so that's not going to work out.

00:00:15.180 --> 00:00:18.810
70 goes into 14-- well, it
doesn't go into 14 any time.

00:00:18.810 --> 00:00:21.270
So we're going to have to
add some decimal places.

00:00:21.270 --> 00:00:22.390
So let's do that.

00:00:22.390 --> 00:00:24.780
Let's add a decimal
place right over here.

00:00:24.780 --> 00:00:26.900
Make sure we put the
decimal right over there.

00:00:26.900 --> 00:00:31.620
But now we can think of it as
70 goes into 140 how many times.

00:00:31.620 --> 00:00:36.880
Well, 70 goes into 14 two times.

00:00:36.880 --> 00:00:41.390
And 2 times 70 is 140.

00:00:41.390 --> 00:00:44.510
And you subtract, and you
are left with no remainder.

00:00:44.510 --> 00:00:45.150
So you're done.

00:00:45.150 --> 00:00:53.370
You can write that 14 divided
by 70 is exactly equal to 0.2.

00:00:53.370 --> 00:00:57.750
Let's do a slightly more
involved example, like this.

00:00:57.750 --> 00:01:05.312
So let's say I wanted to
divide-- oh, I don't know.

00:01:05.312 --> 00:01:06.770
Let's do a little
more interesting.

00:01:06.770 --> 00:01:10.070
Let's divide 16.

00:01:10.070 --> 00:01:14.000
Let's divide 16
by-- I don't know.

00:01:14.000 --> 00:01:18.640
Let's divide it by 25
and see what we get.

00:01:18.640 --> 00:01:20.620
So initially we say, OK,
25 doesn't go into 1.

00:01:20.620 --> 00:01:22.120
It doesn't go into 16.

00:01:22.120 --> 00:01:24.840
We're going to have to add
some decimals right over here.

00:01:24.840 --> 00:01:27.610
How many times does
25 go into 160?

00:01:27.610 --> 00:01:30.160
And make sure we put
the decimal above it.

00:01:30.160 --> 00:01:33.796
So let's see, 25 times 4 is 100.

00:01:33.796 --> 00:01:36.470
25 times 6 is 150,
so that looks right.

00:01:36.470 --> 00:01:38.120
So it goes six times.

00:01:38.120 --> 00:01:44.630
6 times 25 is 120
plus 30, which is 150.

00:01:44.630 --> 00:01:46.530
And now we can subtract.

00:01:46.530 --> 00:01:49.910
And 160 minus 150 is 10.

00:01:49.910 --> 00:01:51.694
And now we just bring
down one more 0.

00:01:51.694 --> 00:01:52.860
We add one more place value.

00:01:52.860 --> 00:01:54.359
Notice we're not
changing the value.

00:01:54.359 --> 00:01:56.730
16 is the same thing as
16.0, which is the same thing

00:01:56.730 --> 00:01:58.580
as 16.00.

00:01:58.580 --> 00:02:01.290
So now you bring down
this 0 right over here.

00:02:01.290 --> 00:02:03.270
And how many times
does 25 go into 100?

00:02:03.270 --> 00:02:05.890
Well, we know it goes
exactly four times.

00:02:05.890 --> 00:02:10.610
4 times 25 is 100,
and we are done.

00:02:10.610 --> 00:02:12.580
We have nothing left
to keep dividing with,

00:02:12.580 --> 00:02:19.470
so we can say that 16 divided
by 25 is equal to 0.64.


Dividing completely to get decimal answer

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:07.660
Let's take 63 and
divide it by 35.

00:00:07.660 --> 00:00:09.320
So the first thing
that we might say

00:00:09.320 --> 00:00:11.970
is, OK, well, 35
doesn't go into 6.

00:00:11.970 --> 00:00:13.205
It does go into 63.

00:00:13.205 --> 00:00:16.850
It goes into 63 one time,
because 2 times 35 is 70,

00:00:16.850 --> 00:00:18.020
so that's too big.

00:00:18.020 --> 00:00:19.710
So it goes one time.

00:00:19.710 --> 00:00:22.010
So let me write that.

00:00:22.010 --> 00:00:24.720
1 times 35 is 35.

00:00:24.720 --> 00:00:28.610
And then if we were to subtract
and we can regroup up here,

00:00:28.610 --> 00:00:31.860
we can take a 10 from the
60, so it becomes a 50,

00:00:31.860 --> 00:00:35.336
give that 10 to the
3, so it becomes a 13.

00:00:35.336 --> 00:00:37.120
13 minus 5 is 8.

00:00:37.120 --> 00:00:39.790
5 minus 3 is 2.

00:00:39.790 --> 00:00:43.050
So you could just say,
hey, 63 divided by 35--

00:00:43.050 --> 00:00:43.800
let me write this.

00:00:43.800 --> 00:00:51.530
You could say 63 divided by
35 is equal to 1 remainder 28.

00:00:54.080 --> 00:00:55.389
But this isn't so satisfying.

00:00:55.389 --> 00:00:56.930
We know that the
real answer is going

00:00:56.930 --> 00:00:59.097
to be one point something,
something, something.

00:00:59.097 --> 00:01:00.680
So what I want to
do is keep dividing.

00:01:00.680 --> 00:01:03.013
I want to divide this thing
completely and see what type

00:01:03.013 --> 00:01:05.110
of a decimal I actually get.

00:01:05.110 --> 00:01:06.820
And to do that, I
essentially have

00:01:06.820 --> 00:01:09.450
to add a decimal here and
then just keep bringing down

00:01:09.450 --> 00:01:12.110
decimal places to the
right of the decimal.

00:01:12.110 --> 00:01:15.630
So 63 is the exact
same thing as 63.0,

00:01:15.630 --> 00:01:19.740
and I could add as many zeroes
as I might want to add here.

00:01:19.740 --> 00:01:21.450
So what we could do
is we just make sure

00:01:21.450 --> 00:01:23.510
that this decimals
right over there,

00:01:23.510 --> 00:01:26.500
and we can now bring down a
zero from the tenths place

00:01:26.500 --> 00:01:27.570
right over here.

00:01:27.570 --> 00:01:30.820
And you bring down that zero,
and now we ask ourselves,

00:01:30.820 --> 00:01:33.696
how many times does
35 go into 280?

00:01:33.696 --> 00:01:35.820
And, as always, this is a
bit of an art when you're

00:01:35.820 --> 00:01:38.680
dividing a two-digit number
into a three-digit number.

00:01:38.680 --> 00:01:40.138
So let's see, it's
definitely going

00:01:40.138 --> 00:01:45.930
to be-- if I were to say-- so
40 goes into 280 seven times.

00:01:45.930 --> 00:01:50.010
30 goes into 280
about nine times.

00:01:50.010 --> 00:01:54.000
It's going to be between
7 and 9, so let's try 8.

00:01:54.000 --> 00:01:56.230
So, let's see what
35 times 8 is.

00:01:56.230 --> 00:01:59.140
35 times 8.

00:01:59.140 --> 00:02:06.230
5 times 8 is 40, 3 times
8 is 24, plus 4 is 28.

00:02:06.230 --> 00:02:08.210
So it actually
works out perfectly.

00:02:08.210 --> 00:02:14.030
So 35 goes into 280
exactly eight times.

00:02:14.030 --> 00:02:16.470
8 times 5, we already
figured it out.

00:02:16.470 --> 00:02:21.745
8 times 35 is exactly 280, and
we don't have any remainder

00:02:21.745 --> 00:02:25.540
now, so we don't have to bring
down any more of these zeroes.

00:02:25.540 --> 00:02:30.880
So now we know exactly
that 63 divided by 35

00:02:30.880 --> 00:02:34.860
is equal to exactly 1.8.


Basic fractional exponents

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:03.570
We already know a good
bit about exponents.

00:00:03.570 --> 00:00:05.930
For example, we know
if we took the number 4

00:00:05.930 --> 00:00:07.770
and raised it to
the third power,

00:00:07.770 --> 00:00:10.590
this is equivalent
to taking three fours

00:00:10.590 --> 00:00:12.040
and multiplying them.

00:00:12.040 --> 00:00:14.670
Or you can also view it
as starting with a 1,

00:00:14.670 --> 00:00:17.600
and then multiplying the 1
by 4, or multiplying that

00:00:17.600 --> 00:00:19.470
by 4, three times.

00:00:19.470 --> 00:00:22.850
But either way, this is
going to result in 4 times 4

00:00:22.850 --> 00:00:26.050
is 16, times 4 is 64.

00:00:26.050 --> 00:00:28.210
We also know a little bit
about negative exponents.

00:00:28.210 --> 00:00:31.810
So for example, if I were take
4 to the negative 3 power,

00:00:31.810 --> 00:00:33.830
we know this negative
tells us to take

00:00:33.830 --> 00:00:38.280
the reciprocal 1/4 to the third.

00:00:38.280 --> 00:00:40.050
And we already know
4 to the third is 64,

00:00:40.050 --> 00:00:43.250
so this is going to be 1/64.

00:00:43.250 --> 00:00:46.470
Now let's think about
fractional exponents.

00:00:46.470 --> 00:00:49.712
So we're going to think about
what is 4 to the 1/2 power.

00:00:49.712 --> 00:00:51.295
And I encourage you
to pause the video

00:00:51.295 --> 00:00:54.550
and at least take a guess
about what you think this is.

00:00:54.550 --> 00:00:57.031
So, the mathematical
convention here,

00:00:57.031 --> 00:00:59.530
the mathematical definition
that most people use, or in fact

00:00:59.530 --> 00:01:03.190
that all people use here,
is that 4 to the 1/2 power

00:01:03.190 --> 00:01:09.200
is the exact same thing
as the square root of 4.

00:01:09.200 --> 00:01:11.190
And we'll talk in the
future about why this is,

00:01:11.190 --> 00:01:13.023
and the reason why this
is defined this way,

00:01:13.023 --> 00:01:15.760
is it has all sorts of
neat and elegant properties

00:01:15.760 --> 00:01:20.090
when you start manipulating
the actual exponents.

00:01:20.090 --> 00:01:21.590
But what is the
square root of 4,

00:01:21.590 --> 00:01:23.950
especially the
principal root, mean?

00:01:23.950 --> 00:01:26.950
Well that means,
well, what is a number

00:01:26.950 --> 00:01:29.362
that if I were to
multiply it by itself,

00:01:29.362 --> 00:01:31.070
or if I were to have
two of those numbers

00:01:31.070 --> 00:01:33.310
and I were to multiply
them, times each other,

00:01:33.310 --> 00:01:35.240
that same number,
I'm going to get 4?

00:01:35.240 --> 00:01:39.210
Well, what times
itself is equal to 4?

00:01:39.210 --> 00:01:43.150
Well that's of
course equal to 2.

00:01:43.150 --> 00:01:46.560
And just to get a sense of why
this starts to work out, well

00:01:46.560 --> 00:01:48.460
remember, we could
have also written

00:01:48.460 --> 00:01:53.400
that 4 is equal to 2 squared.

00:01:53.400 --> 00:01:55.400
So you're starting to see
something interesting.

00:01:55.400 --> 00:02:02.420
4 to the 1/2 is equal to
2, 2 squared is equal to 4.

00:02:02.420 --> 00:02:04.310
So let's get a couple
more examples of this,

00:02:04.310 --> 00:02:07.110
just so you make sure
you get what's going on.

00:02:07.110 --> 00:02:09.515
And I encourage you to pause
it as much as necessary

00:02:09.515 --> 00:02:11.370
and try to figure
it out yourself.

00:02:11.370 --> 00:02:13.130
So based on what
I just told you,

00:02:13.130 --> 00:02:17.385
what do you think 9 to the
1/2 power is going to be?

00:02:17.385 --> 00:02:19.010
Well, that's just
the square root of 9.

00:02:19.010 --> 00:02:22.412
The principal root of 9, that's
just going to be equal to 3.

00:02:22.412 --> 00:02:23.870
And likewise, we
could've also said

00:02:23.870 --> 00:02:26.830
that 3 squared is, or
let me write it this way,

00:02:26.830 --> 00:02:31.160
that 9 is equal to 3 squared.

00:02:31.160 --> 00:02:35.150
These are both true statements.

00:02:35.150 --> 00:02:36.350
Let's do one more like this.

00:02:36.350 --> 00:02:39.900
What is 25 to the
1/2 going to be?

00:02:39.900 --> 00:02:41.430
Well, this is just
going to be 5.

00:02:41.430 --> 00:02:42.655
5 times 5 is 25.

00:02:42.655 --> 00:02:47.602
Or you could say, 25
is equal to 5 squared.

00:02:47.602 --> 00:02:50.060
Now, let's think about what
happens when you take something

00:02:50.060 --> 00:02:52.310
to the 1/3 power.

00:02:52.310 --> 00:02:58.280
So let's imagine taking
8 to the 1/3 power.

00:02:58.280 --> 00:03:00.380
So the definition here
is that taking something

00:03:00.380 --> 00:03:02.270
to the 1/3 power
is the same thing

00:03:02.270 --> 00:03:09.535
as taking the cube
root of that number.

00:03:09.535 --> 00:03:11.660
And the cube root is just
saying, well what number,

00:03:11.660 --> 00:03:14.390
if I had three of that
number, and I multiply them,

00:03:14.390 --> 00:03:15.970
that I'm going to get 8.

00:03:15.970 --> 00:03:18.990
So something, times something,
times something, is 8.

00:03:18.990 --> 00:03:23.980
Well, we already know that 8 is
equal to 2 to the third power.

00:03:23.980 --> 00:03:26.690
So the cube root of
8, or 8 to the 1/3,

00:03:26.690 --> 00:03:28.176
is just going to be equal to 2.

00:03:28.176 --> 00:03:29.550
This says hey,
give me the number

00:03:29.550 --> 00:03:33.680
that if I say that number, times
that number, times that number,

00:03:33.680 --> 00:03:35.080
I'm going to get 8.

00:03:35.080 --> 00:03:38.490
Well, that number is 2 because
2 to the third power is 8.

00:03:38.490 --> 00:03:40.250
Do a few more examples of that.

00:03:40.250 --> 00:03:44.220
What is 64 to the 1/3 power?

00:03:44.220 --> 00:03:48.390
Well, we already know that
4 times 4 times 4 is 64.

00:03:48.390 --> 00:03:52.100
So this is going to be 4.

00:03:52.100 --> 00:03:55.285
And we already wrote over here
that 64 is the same thing as 4

00:03:55.285 --> 00:03:56.027
to the third.

00:03:56.027 --> 00:03:58.610
I think you're starting to see
a little bit of a pattern here,

00:03:58.610 --> 00:04:00.650
a little bit of symmetry here.

00:04:00.650 --> 00:04:06.370
And we can extend this idea to
arbitrary rational exponents.

00:04:06.370 --> 00:04:12.630
So what happens if I were
to raise-- let's say I had,

00:04:12.630 --> 00:04:17.240
let me think of a good number
here-- so let's say I have 32.

00:04:17.240 --> 00:04:24.451
I have the number 32, and I
raise it to the 1/5 power.

00:04:24.451 --> 00:04:25.950
So this says hey,
give me the number

00:04:25.950 --> 00:04:28.450
that if I were to
multiply that number,

00:04:28.450 --> 00:04:31.360
or I were to repeatedly
multiply that number five times,

00:04:31.360 --> 00:04:34.460
what is that, I would get 32.

00:04:34.460 --> 00:04:40.270
Well, 32 is the same thing as
2 times 2 times 2 times 2 times

00:04:40.270 --> 00:04:41.170
2.

00:04:41.170 --> 00:04:43.920
So 2 is that number, that
if I were to multiply it

00:04:43.920 --> 00:04:47.150
five times, then
I'm going to get 32.

00:04:47.150 --> 00:04:49.770
So this right over here
is 2, or another way

00:04:49.770 --> 00:04:52.790
of saying this kind of same
statement about the world

00:04:52.790 --> 00:04:58.280
is that 32 is equal to
2 to the fifth power.


Negative fractional exponent examples

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:02.950
Let's do some slightly
more complicated

00:00:02.950 --> 00:00:05.110
fractional exponent examples.

00:00:05.110 --> 00:00:06.690
So we already know
that if I were

00:00:06.690 --> 00:00:11.910
to take 9 to the 1/2 power,
this is going to be equal to 3,

00:00:11.910 --> 00:00:14.030
and we know that because
3 times 3 is equal to 9.

00:00:14.030 --> 00:00:15.446
This is equivalent
to saying, what

00:00:15.446 --> 00:00:16.890
is the principal root of 9?

00:00:16.890 --> 00:00:19.020
Well, that is equal to 3.

00:00:19.020 --> 00:00:23.400
But what would happen if I took
9 to the negative 1/2 power?

00:00:23.400 --> 00:00:26.010
Now we have a negative
fractional exponent,

00:00:26.010 --> 00:00:27.600
and the key to
this is to just not

00:00:27.600 --> 00:00:30.630
get too worried or
intimidated by this,

00:00:30.630 --> 00:00:32.210
but just think about
it step by step.

00:00:32.210 --> 00:00:34.709
Just ignore for the second
that this is a fraction,

00:00:34.709 --> 00:00:36.250
and just look at
this negative first.

00:00:36.250 --> 00:00:37.980
Just breathe
slowly, and realize,

00:00:37.980 --> 00:00:40.630
OK, I got a negative exponent.

00:00:40.630 --> 00:00:44.190
That means that this is
just going to be 1 over 9

00:00:44.190 --> 00:00:45.860
to the 1/2.

00:00:45.860 --> 00:00:48.380
That's what that
negative is a cue for.

00:00:48.380 --> 00:00:53.880
This is 1 over 9 to the 1/2,
and we know that 9 to the 1/2

00:00:53.880 --> 00:00:56.200
is equal to 3.

00:00:56.200 --> 00:00:59.165
So this is just going
to be equal to 1/3.

00:01:03.420 --> 00:01:06.180
Let's take things a
little bit further.

00:01:06.180 --> 00:01:07.970
What would this evaluate to?

00:01:07.970 --> 00:01:10.680
And I encourage you to pause
the video after trying it,

00:01:10.680 --> 00:01:12.660
or pause the video to try it.

00:01:12.660 --> 00:01:21.539
Negative 27 to the
negative 1/3 power.

00:01:21.539 --> 00:01:23.080
So I encourage you
to pause the video

00:01:23.080 --> 00:01:26.160
and think about what
this would evaluate to.

00:01:26.160 --> 00:01:27.820
So remember, just
take a deep breath.

00:01:27.820 --> 00:01:30.111
You can always get rid of
this negative in the exponent

00:01:30.111 --> 00:01:32.590
by taking the reciprocal and
raising it to the positive.

00:01:32.590 --> 00:01:38.340
So this is going to be
equal to 1 over negative 27

00:01:38.340 --> 00:01:40.330
to the positive 1/3 power.

00:01:40.330 --> 00:01:41.580
And I know what you're saying.

00:01:41.580 --> 00:01:43.020
Hey, I still can't
breathe easily.

00:01:43.020 --> 00:01:46.250
I have this negative number
to this fractional exponent.

00:01:46.250 --> 00:01:48.700
But this is just
saying what number,

00:01:48.700 --> 00:01:51.570
if I were to multiply
it three times--

00:01:51.570 --> 00:01:54.200
so if I have that number, so
whatever the number this is,

00:01:54.200 --> 00:01:56.820
if I were to multiply it,
if I took three of them

00:01:56.820 --> 00:01:59.050
and I multiply them
together, if I multiplied 1

00:01:59.050 --> 00:02:03.010
by that number three
times, what number would I

00:02:03.010 --> 00:02:06.930
have to use here
to get negative 27?

00:02:06.930 --> 00:02:11.140
Well, we already know
that 3 to the third,

00:02:11.140 --> 00:02:14.130
which is equal to
3 times 3 times 3,

00:02:14.130 --> 00:02:15.890
is equal to positive 27.

00:02:15.890 --> 00:02:18.080
So that's a pretty good clue.

00:02:18.080 --> 00:02:21.770
What would negative 3
to the third power be?

00:02:21.770 --> 00:02:27.210
Well, that's negative 3 times
negative 3 times negative 3,

00:02:27.210 --> 00:02:30.130
which is negative 3 times
negative 3 is positive 9.

00:02:30.130 --> 00:02:33.104
Times negative 3 is negative 27.

00:02:33.104 --> 00:02:35.270
So we've just found this
number, this question mark.

00:02:35.270 --> 00:02:37.390
Negative 3 times negative
3 times negative 3

00:02:37.390 --> 00:02:39.300
is equal to negative 27.

00:02:39.300 --> 00:02:42.910
So negative 27 to the
1/3-- this part right over

00:02:42.910 --> 00:02:45.660
here-- is equal to negative 3.

00:02:45.660 --> 00:02:54.080
So this is going to be
equal to 1 over negative 3,

00:02:54.080 --> 00:03:00.460
which is the same
thing as negative 1/3.


Evaluating fractional exponents: fractional base

WEBVTT
Kind: captions
Language: en

00:00:00.309 --> 00:00:02.600
Fractional exponents can be
a little daunting at first,

00:00:02.600 --> 00:00:04.810
so it never hurts to do as
many examples as possible.

00:00:04.810 --> 00:00:06.150
So let's do a few.

00:00:06.150 --> 00:00:11.720
What if we had
25/9, and we wanted

00:00:11.720 --> 00:00:14.885
to raise it to the 1/2 power?

00:00:14.885 --> 00:00:16.510
So we're essentially
just saying, well,

00:00:16.510 --> 00:00:21.730
what is the principal
square root of 25/9?

00:00:21.730 --> 00:00:26.110
So what number times
itself is going to be 25/9?

00:00:26.110 --> 00:00:30.710
Well, we know 5 times 5
is 25, and 3 times 3 is 9.

00:00:30.710 --> 00:00:33.520
So why don't we
just go with 5/3?

00:00:33.520 --> 00:00:39.560
Because notice, if you
have 5/3 times 5/3,

00:00:39.560 --> 00:00:42.850
that is going to be 25/9.

00:00:42.850 --> 00:00:45.740
Or another way of saying
this, that 5/3 squared

00:00:45.740 --> 00:00:48.170
is equal to 25/9.

00:00:48.170 --> 00:00:54.200
So 25/9 to the 1/2 is
going to be equal to 5/3.

00:00:54.200 --> 00:00:58.330
Now let's escalate
things a little bit.

00:00:58.330 --> 00:01:00.310
Let's take a really hairy one.

00:01:00.310 --> 00:01:17.690
Let's raise 81/256 to
the negative 1/4 power.

00:01:17.690 --> 00:01:21.211
I encourage you to pause this
and try this on your own.

00:01:21.211 --> 00:01:22.210
So what's going on here?

00:01:22.210 --> 00:01:24.335
This negative-- the first
thing I always like to do

00:01:24.335 --> 00:01:26.970
is I want to get rid of this
negative in the exponent.

00:01:26.970 --> 00:01:29.530
So let me just take
the reciprocal of this

00:01:29.530 --> 00:01:31.430
and raise it to the positive.

00:01:31.430 --> 00:01:37.220
So I could just say that
this is equal to 256/81

00:01:37.220 --> 00:01:40.050
to the 1/4 power.

00:01:40.050 --> 00:01:42.210
And so now I can say,
well, what number

00:01:42.210 --> 00:01:45.167
times itself times
itself times itself

00:01:45.167 --> 00:01:47.000
is going to be equal
to 256, and what number

00:01:47.000 --> 00:01:48.960
times itself times
itself times itself--

00:01:48.960 --> 00:01:50.410
did I say that four times?

00:01:50.410 --> 00:01:53.200
Well, what number, if I take
four of them and multiply,

00:01:53.200 --> 00:01:55.072
do I get 81?

00:01:55.072 --> 00:01:56.530
And one way to
think about it, this

00:01:56.530 --> 00:01:57.800
is going to be the
same thing-- and we'll

00:01:57.800 --> 00:01:59.270
talk about this in
more depth later

00:01:59.270 --> 00:02:01.260
on when we talk about
exponent properties.

00:02:01.260 --> 00:02:04.420
But this is going to be
the exact same thing as 256

00:02:04.420 --> 00:02:08.309
to the 1/4 over 81 to the 1/4.

00:02:08.309 --> 00:02:09.600
You, in fact, saw it over here.

00:02:09.600 --> 00:02:12.550
This over here was the same
thing as the square root of 25

00:02:12.550 --> 00:02:14.120
over the square root of 9.

00:02:14.120 --> 00:02:17.852
Or 25 to the 1/2
over 9 to the 1/2.

00:02:17.852 --> 00:02:19.310
So we're just doing
that over here.

00:02:19.310 --> 00:02:22.170
So one, we still have to think
about what number this is.

00:02:22.170 --> 00:02:24.670
And this is a little bit of,
there's no easy way to do this.

00:02:24.670 --> 00:02:25.740
You kind of have
to just play around

00:02:25.740 --> 00:02:27.250
a little bit to come up with it.

00:02:27.250 --> 00:02:30.650
But 4 might jump out at you if
you recognize that 16 times 16

00:02:30.650 --> 00:02:31.930
is 256.

00:02:31.930 --> 00:02:34.100
We know that 4 to
the fourth power,

00:02:34.100 --> 00:02:38.750
or you're about to know this,
is 4 times 4 times 4 times 4.

00:02:38.750 --> 00:02:44.490
And 4 times 4 is 16, times 4
is 64, times 4 is equal to 256.

00:02:44.490 --> 00:02:47.270
So 4 to the fourth
is 256, or we could

00:02:47.270 --> 00:02:51.360
say 4 is equal to
256 to the 1/4 power.

00:02:51.360 --> 00:02:52.230
Fair enough?

00:02:52.230 --> 00:02:53.980
Now what about 81?

00:02:53.980 --> 00:02:56.020
Well, 3 might jump out at you.

00:02:56.020 --> 00:02:58.190
We know that 3 to
the fourth power

00:02:58.190 --> 00:03:04.840
is equal to 3 times 3 times 3
times 3, which is equal to 81.

00:03:04.840 --> 00:03:08.730
So 3 is equal to 81 to the 1/4.

00:03:08.730 --> 00:03:13.916
So this top number, 256
to the 1/4, that's just 4.

00:03:13.916 --> 00:03:17.640
81 to the 1/4, that is just 3.

00:03:17.640 --> 00:03:22.590
So this right over here is
going to be equal to 4/3.


Slope of a secant line example 3

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:02.290
Consider the graph
of the function

00:00:02.290 --> 00:00:05.220
f of x that passes through
three points as shown.

00:00:05.220 --> 00:00:07.800
So these are the three
points, and this curve in blue

00:00:07.800 --> 00:00:09.280
is f of x.

00:00:09.280 --> 00:00:12.610
Identify which of the
statements are true.

00:00:12.610 --> 00:00:14.030
So they give us
these statements.

00:00:14.030 --> 00:00:14.530
Let's see.

00:00:14.530 --> 00:00:18.280
This first one says f of
negative a is less than 1

00:00:18.280 --> 00:00:19.764
minus f of negative a over a.

00:00:19.764 --> 00:00:21.930
So this seems like some
type of a bizarre statement.

00:00:21.930 --> 00:00:24.013
How are we able to figure
out whether this is true

00:00:24.013 --> 00:00:25.620
from this right over here?

00:00:25.620 --> 00:00:27.290
So let's just go
piece by piece and see

00:00:27.290 --> 00:00:29.420
if something starts
to make sense.

00:00:29.420 --> 00:00:31.990
So f of negative a, where
do we see that here?

00:00:31.990 --> 00:00:33.590
Well, this is f of negative a.

00:00:33.590 --> 00:00:36.750
This is the point x
equals negative a.

00:00:36.750 --> 00:00:39.460
So this is negative
a, and this is

00:00:39.460 --> 00:00:42.220
y is equal to f of negative a.

00:00:42.220 --> 00:00:46.450
So this is f of negative
a right over here.

00:00:46.450 --> 00:00:48.600
And what we know about
f of negative a, based

00:00:48.600 --> 00:00:50.710
on looking at this graph,
is that f of negative a

00:00:50.710 --> 00:00:52.675
is between 0 and 1.

00:00:52.675 --> 00:00:53.550
So we can write that.

00:00:53.550 --> 00:00:58.580
0 is less than f of negative
a, which is less than 1.

00:00:58.580 --> 00:01:02.060
So that's all I can deduce
about f of negative a right

00:01:02.060 --> 00:01:03.740
from the get-go.

00:01:03.740 --> 00:01:05.840
Now let's look at this
crazy statement, 1

00:01:05.840 --> 00:01:10.970
minus f of negative a over a.

00:01:10.970 --> 00:01:12.281
What is this?

00:01:12.281 --> 00:01:13.780
Well, let's think
about what happens

00:01:13.780 --> 00:01:15.770
if we take the
secant line, if we're

00:01:15.770 --> 00:01:17.940
trying to find the slope
of the secant line,

00:01:17.940 --> 00:01:20.272
between this point
and this point,

00:01:20.272 --> 00:01:22.230
if we wanted to find the
average rate of change

00:01:22.230 --> 00:01:24.990
between the point negative a,
f of negative a, and the point

00:01:24.990 --> 00:01:25.760
0, 1.

00:01:25.760 --> 00:01:30.690
If this is our endpoint,
our change in y

00:01:30.690 --> 00:01:34.340
is going to be 1
minus f of negative a.

00:01:34.340 --> 00:01:41.130
So 1 minus f of negative a
is equal to our change in y.

00:01:41.130 --> 00:01:48.230
And our change in x, going from
negative a to 0, so change in x

00:01:48.230 --> 00:01:52.120
is going to be equal to
0 minus negative a, which

00:01:52.120 --> 00:01:54.340
is equal to positive a.

00:01:54.340 --> 00:01:56.540
So this right over here
is essentially our change

00:01:56.540 --> 00:01:59.320
in x over our change in y
from this point to this point.

00:01:59.320 --> 00:02:04.170
It is our average rate of change
from this point to this point.

00:02:04.170 --> 00:02:06.196
So it is our average
rate of change,

00:02:06.196 --> 00:02:08.320
or you could say it's the
slope of the secant line.

00:02:08.320 --> 00:02:10.403
So the secant line would
look something like this.

00:02:15.019 --> 00:02:17.060
Slope of the secant line,
so this right over here

00:02:17.060 --> 00:02:28.350
is slope of secant
line between from f

00:02:28.350 --> 00:02:33.600
of negative a to 0 comma 1.

00:02:33.600 --> 00:02:36.090
So just looking at this
diagram right over here,

00:02:36.090 --> 00:02:38.175
what do we know
about this slope?

00:02:38.175 --> 00:02:40.050
And in particular, can
we make any statements

00:02:40.050 --> 00:02:43.290
about that slope
relative to, say, 0 or 1

00:02:43.290 --> 00:02:45.000
or anything like that?

00:02:45.000 --> 00:02:47.730
Well, let's think about
what a line of slope 1

00:02:47.730 --> 00:02:49.000
would look like.

00:02:49.000 --> 00:02:50.590
Well, a line of
slope 1, especially

00:02:50.590 --> 00:02:52.631
one that went through this
point right over here,

00:02:52.631 --> 00:02:56.310
would look something like this.

00:02:56.310 --> 00:03:00.639
A line of slope 1 would
look something like this.

00:03:00.639 --> 00:03:02.430
So this line right over
here that I've just

00:03:02.430 --> 00:03:06.330
drawn that goes from
negative 1 comma 0 to 0, 1,

00:03:06.330 --> 00:03:07.480
this has slope 1.

00:03:07.480 --> 00:03:10.550
So this slope is equal to 1.

00:03:10.550 --> 00:03:12.860
So if this green line
has a slope of 1,

00:03:12.860 --> 00:03:14.510
does this blue
line have a slope--

00:03:14.510 --> 00:03:16.160
it clearly has a
different slope.

00:03:16.160 --> 00:03:18.960
Is that slope, is
this blue line steeper

00:03:18.960 --> 00:03:21.370
or less steep than
the green line?

00:03:21.370 --> 00:03:23.570
Well, it's pretty clear
that this secant line

00:03:23.570 --> 00:03:25.930
is steeper than the green line.

00:03:25.930 --> 00:03:27.570
It's increasing faster.

00:03:27.570 --> 00:03:30.290
So it's going to
have a higher slope.

00:03:30.290 --> 00:03:32.850
So this, looking at
it from this diagram,

00:03:32.850 --> 00:03:36.820
this blue line has a
slope higher than 1.

00:03:36.820 --> 00:03:39.135
Or the slope of the secant
line from negative a, f

00:03:39.135 --> 00:03:41.090
of negative a, to
0 comma 1, that

00:03:41.090 --> 00:03:43.610
is going to be greater than 1.

00:03:43.610 --> 00:03:46.215
So this thing right over
here is greater than 1.

00:03:46.215 --> 00:03:48.340
So we're able to deduce,
this thing right over here

00:03:48.340 --> 00:03:49.640
is less than 1.

00:03:49.640 --> 00:03:52.420
This thing over here
is greater than 1.

00:03:52.420 --> 00:03:54.500
So this thing is
less than that thing.

00:03:54.500 --> 00:03:57.565
So this must be true.

00:03:57.565 --> 00:03:58.690
Now let's look at this one.

00:03:58.690 --> 00:04:01.337
We're comparing the
slope of the secant line.

00:04:01.337 --> 00:04:03.170
We're comparing the
slope of the secant line

00:04:03.170 --> 00:04:04.191
that we just looked at.

00:04:04.191 --> 00:04:05.940
So this is the same
value right over here.

00:04:05.940 --> 00:04:09.020
So we're comparing this
slope right over here,

00:04:09.020 --> 00:04:12.400
to what's this? f
of a minus 1 over a.

00:04:12.400 --> 00:04:15.600
Well, this is the slope
of this secant line,

00:04:15.600 --> 00:04:17.510
this is the slope
of this secant line

00:04:17.510 --> 00:04:20.000
that I'm drawing in this--
let me do it in more contrast.

00:04:20.000 --> 00:04:21.920
Let me do it in orange.

00:04:21.920 --> 00:04:28.830
That is the slope
of this secant line.

00:04:28.830 --> 00:04:32.040
So which one has a higher slope?

00:04:32.040 --> 00:04:34.230
Well, it's pretty clear
that the blue secant line

00:04:34.230 --> 00:04:37.730
has a higher slope than
this orange secant line.

00:04:37.730 --> 00:04:39.960
But here, it's saying
that the blue's slope

00:04:39.960 --> 00:04:41.380
is lower than the orange.

00:04:41.380 --> 00:04:43.450
So this is not going to be true.

00:04:43.450 --> 00:04:45.690
So this is not true.

00:04:45.690 --> 00:04:49.370
Then finally, let's look at
this over here-- f of a minus f

00:04:49.370 --> 00:04:52.770
of negative a over 2a.

00:04:52.770 --> 00:04:55.800
So this is the slope.

00:04:55.800 --> 00:04:57.160
Let me draw this.

00:04:57.160 --> 00:04:59.910
So this right over
here, this is the slope

00:04:59.910 --> 00:05:03.130
of the secant line
between this point

00:05:03.130 --> 00:05:04.740
and this point right over here.

00:05:04.740 --> 00:05:07.820
Our change in y is f of
a minus f of negative a.

00:05:07.820 --> 00:05:11.050
Our change in x is a minus
negative a, which is 2a.

00:05:11.050 --> 00:05:13.460
So this is this secant
line right over here.

00:05:13.460 --> 00:05:15.310
So let me draw it.

00:05:15.310 --> 00:05:19.740
So this secant line
right over here,

00:05:19.740 --> 00:05:25.930
so they're comparing
that slope to this slope.

00:05:25.930 --> 00:05:29.520
f of a minus 1 is our change in
y, over a is our change in x.

00:05:29.520 --> 00:05:32.049
So we're comparing it to
that one right over there.

00:05:32.049 --> 00:05:33.840
And you could immediately
eyeball this kind

00:05:33.840 --> 00:05:35.600
of brownish,
maroon-- I guess it's

00:05:35.600 --> 00:05:39.370
kind of a brown color-- this
secant line that goes all

00:05:39.370 --> 00:05:42.730
the way from here to here is
clearly steeper than this one

00:05:42.730 --> 00:05:43.440
right over here.

00:05:43.440 --> 00:05:45.481
And we know that, that
the average rate of change

00:05:45.481 --> 00:05:47.210
from here to here is
going to be higher

00:05:47.210 --> 00:05:49.330
than the average rate of change
from here to here, because

00:05:49.330 --> 00:05:50.890
at least from
negative a to 0, we

00:05:50.890 --> 00:05:52.910
were increasing at
a much faster rate.

00:05:52.910 --> 00:05:54.590
And then we slowed
down to this rate.

00:05:54.590 --> 00:05:56.560
So the average over
the entire interval

00:05:56.560 --> 00:06:02.420
is definitely going to be more
than what we get from 0 to a.

00:06:02.420 --> 00:06:04.640
So this one is also not true.

00:06:04.640 --> 00:06:08.204
This has a higher-- we actually
know that this is false.

00:06:08.204 --> 00:06:09.620
Both of these would
have been true

00:06:09.620 --> 00:06:11.254
if we swapped
these signs around,

00:06:11.254 --> 00:06:12.920
if this was a greater
than sign, if this

00:06:12.920 --> 00:06:14.270
was a greater than sign.

00:06:14.270 --> 00:06:17.480
So this is the only
one that applies.


Negative exponents

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:05.100
We already know that
2 to the fourth power

00:00:05.100 --> 00:00:08.200
can be viewed as
starting with a 1

00:00:08.200 --> 00:00:11.450
and then multiplying
it by 2 four times.

00:00:11.450 --> 00:00:12.300
So let me do that.

00:00:12.300 --> 00:00:18.100
So times 2, times
2, times 2, times 2.

00:00:18.100 --> 00:00:22.010
And that will give us, let's
see, 2 times 2 is 4, 8, 16.

00:00:22.010 --> 00:00:24.220
So that will give us 16.

00:00:24.220 --> 00:00:27.080
Now I will ask you a more
interesting question.

00:00:27.080 --> 00:00:33.080
What do you think 2 to
the negative 4 power is?

00:00:33.080 --> 00:00:37.659
And I encourage you to pause
the video and think about that.

00:00:37.659 --> 00:00:39.450
Well, you might be
tempted to say, oh maybe

00:00:39.450 --> 00:00:41.370
it's negative 16 or
something like that,

00:00:41.370 --> 00:00:44.929
but remember what the exponent
operation is trying to do.

00:00:44.929 --> 00:00:46.720
One way of viewing it
is this is telling us

00:00:46.720 --> 00:00:50.770
how many times are we going to
multiply 2 times negative 1?

00:00:50.770 --> 00:00:53.790
But here we're going to
multiply negative 4 times.

00:00:53.790 --> 00:00:55.850
Well, what does negative
traditionally mean?

00:00:55.850 --> 00:00:58.190
Negative traditionally
means the opposite.

00:00:58.190 --> 00:01:00.840
So here this is how many times
you're going to multiply.

00:01:00.840 --> 00:01:02.640
Maybe when we make it
negative this says,

00:01:02.640 --> 00:01:04.306
how many times are
we going to, starting

00:01:04.306 --> 00:01:08.474
with the 1, how many times
are we going to divide by 2?

00:01:08.474 --> 00:01:10.140
So let's think about
that a little that.

00:01:10.140 --> 00:01:13.040
So this could be
viewed as 1 times,

00:01:13.040 --> 00:01:15.580
and we're going to
divide by 2 four times.

00:01:15.580 --> 00:01:19.210
Well, dividing by 2 is the same
thing as multiplying by 1/2.

00:01:19.210 --> 00:01:25.090
So we could say that
this is 1 times 1/2,

00:01:25.090 --> 00:01:27.930
times-- let me just
do it in one color.

00:01:27.930 --> 00:01:34.000
So 1 times 1/2, times
1/2, times 1/2, times 1/2.

00:01:34.000 --> 00:01:35.960
Notice multiplying
by 1/2 four times

00:01:35.960 --> 00:01:40.000
is the exact same thing as
dividing by 2 four times.

00:01:40.000 --> 00:01:43.010
And in this situation
this would get you,

00:01:43.010 --> 00:01:46.820
well 1/2, well 1 times 1/2 half
is just 1/2, times 1/2 is 1/4,

00:01:46.820 --> 00:01:52.200
times 1/2 is 1/8,
times 1/2 is 1/16.

00:01:52.200 --> 00:01:54.484
And so you probably see
the relationship here.

00:01:54.484 --> 00:01:56.900
If you're-- this is essentially
you're starting with the 1

00:01:56.900 --> 00:01:59.310
and you're dividing
by 2 four times.

00:01:59.310 --> 00:02:06.160
You could also say that 2-- I'm
going to do the same colors-- 2

00:02:06.160 --> 00:02:14.150
to the negative 4 is
the same thing as 1/2

00:02:14.150 --> 00:02:15.110
to the fourth power.

00:02:18.060 --> 00:02:19.730
Let me color code
it nicely so you

00:02:19.730 --> 00:02:21.700
realize what the
negative is doing.

00:02:21.700 --> 00:02:23.117
So this negative
right over here--

00:02:23.117 --> 00:02:24.491
let me do that in
a better color,

00:02:24.491 --> 00:02:26.537
I'll do it in magenta,
something that jumps out.

00:02:26.537 --> 00:02:28.120
So this negative
right over here, this

00:02:28.120 --> 00:02:30.680
is what's causing
us to go one over.

00:02:30.680 --> 00:02:33.750
So 2 to the negative
4 is the same thing,

00:02:33.750 --> 00:02:39.590
based on the way we've
defined it just up right here,

00:02:39.590 --> 00:02:43.320
as reciprocal of 2 to
the fourth, or 1 over 2

00:02:43.320 --> 00:02:44.330
to the fourth.

00:02:44.330 --> 00:02:51.922
And so you could view this as
being 1/2 times 2 times 2 times

00:02:51.922 --> 00:02:53.380
2, if you just view
2 to the fourth

00:02:53.380 --> 00:02:55.470
as taking four 2's
and multiplying them.

00:02:55.470 --> 00:02:57.240
Or if you use this
idea right over here,

00:02:57.240 --> 00:02:59.710
you could view it
as starting with a 1

00:02:59.710 --> 00:03:02.770
and multiplying it
by 2 four times.

00:03:02.770 --> 00:03:05.270
Either way, you are
going to get 1/16.

00:03:10.440 --> 00:03:12.310
So let's do a few
more examples of this

00:03:12.310 --> 00:03:15.290
just so that we make sure
things are clear to us.

00:03:15.290 --> 00:03:24.347
So let's try 3 to the
negative third power.

00:03:24.347 --> 00:03:26.180
So remember, whenever
you see that negative,

00:03:26.180 --> 00:03:27.693
what my brain
always does is say I

00:03:27.693 --> 00:03:29.200
need to take the
reciprocal here.

00:03:29.200 --> 00:03:31.200
So this is going
to be equal to, I'm

00:03:31.200 --> 00:03:33.180
going to highlight
the negative again,

00:03:33.180 --> 00:03:36.700
this is going to be 1
over 3 to the third power.

00:03:39.800 --> 00:03:47.520
Which would be equal to 1/3
times 3 times 3, or 1 times

00:03:47.520 --> 00:03:50.760
3 times 3 times 3,
is going to be 27.

00:03:50.760 --> 00:03:54.280
So this is going to be 1/27.

00:03:54.280 --> 00:03:57.150
Let's try another example,
I'll do two or three more.

00:03:57.150 --> 00:04:00.280
So let's take a negative number
to a negative exponent, just

00:04:00.280 --> 00:04:02.350
to see if we can
confuse ourselves.

00:04:02.350 --> 00:04:07.457
So let's take the
number negative 4,

00:04:07.457 --> 00:04:10.040
and let's take it-- I don't want
my numbers to get too big too

00:04:10.040 --> 00:04:10.540
fast.

00:04:10.540 --> 00:04:16.279
So let's just take
negative 2 and let's

00:04:16.279 --> 00:04:20.290
take it to the negative 3 power.

00:04:20.290 --> 00:04:23.840
I'll make my negatives in
magenta, negative 3 power.

00:04:29.200 --> 00:04:30.750
So at first this
might be daunting,

00:04:30.750 --> 00:04:31.750
do the negatives cancel?

00:04:31.750 --> 00:04:33.770
And that will just be the
remnants in your brain

00:04:33.770 --> 00:04:35.980
that are trying to think
of multiplying negatives.

00:04:35.980 --> 00:04:37.640
Do not apply that here.

00:04:37.640 --> 00:04:39.590
Remember, you see a
negative exponent,

00:04:39.590 --> 00:04:43.230
that just means the reciprocal
of the positive exponent.

00:04:43.230 --> 00:04:49.670
So 1 over negative 2
to the third power,

00:04:49.670 --> 00:04:51.600
to the positive third power.

00:04:51.600 --> 00:04:59.370
And this is equal to 1
over negative 2 times

00:04:59.370 --> 00:05:03.640
negative 2 times negative 2.

00:05:03.640 --> 00:05:05.980
Or you could view it as
1 times negative 2 times

00:05:05.980 --> 00:05:07.850
negative 2 times
negative 2, which

00:05:07.850 --> 00:05:13.500
is going to give you 1 over
negative 8 or negative 1/8.

00:05:13.500 --> 00:05:14.935
Let me scroll over
a little bit, I

00:05:14.935 --> 00:05:16.810
don't want to have to
start squishing things.

00:05:16.810 --> 00:05:19.850
So this is equal
to negative 1/8.

00:05:19.850 --> 00:05:21.860
Let's do one more example,
just in an attempt

00:05:21.860 --> 00:05:23.750
to confuse ourselves.

00:05:23.750 --> 00:05:42.320
Let's take 5/8 and raise
this to the negative 2 power.

00:05:42.320 --> 00:05:44.490
So once again, this negative,
oh I got at a fraction

00:05:44.490 --> 00:05:45.290
is a negative here.

00:05:45.290 --> 00:05:49.900
Remember this just means 1
over 5/8 to the second power.

00:05:49.900 --> 00:05:52.080
So this is just going
to be the same thing

00:05:52.080 --> 00:06:03.630
as 1 over 5/8 squared,
which is going

00:06:03.630 --> 00:06:09.230
to be the same thing-- so this
is going to be equal to-- I'm

00:06:09.230 --> 00:06:14.173
trying to color code it, 1 over
5/8 times 5/8, which is 25/64.

00:06:18.890 --> 00:06:22.100
1 over 25/64 is just
going to be 64/25.

00:06:24.630 --> 00:06:27.070
So another way to
think about it is,

00:06:27.070 --> 00:06:29.320
you're going to take
the reciprocal of this

00:06:29.320 --> 00:06:33.010
and raise it to the
positive exponent.

00:06:33.010 --> 00:06:36.940
So another way you could
have thought about this

00:06:36.940 --> 00:06:39.900
is 5/8 to the negative 2 power.

00:06:39.900 --> 00:06:44.090
Let me just take the
reciprocal of this, 8/5

00:06:44.090 --> 00:06:46.670
and raise it to the
positive 2 power.

00:06:46.670 --> 00:06:48.660
So all of these
statements are equivalent.

00:06:48.660 --> 00:06:50.201
And that would have
applied even when

00:06:50.201 --> 00:06:52.260
you're dealing with
non-fractions as your base

00:06:52.260 --> 00:06:53.460
right over here.

00:06:53.460 --> 00:06:56.410
So 2, you could say well this
is going to be the same thing.

00:06:56.410 --> 00:06:57.840
2 to the negative
4 is going to be

00:06:57.840 --> 00:07:00.579
the same thing as
taking my reciprocal.

00:07:00.579 --> 00:07:02.120
So this is going to
be the same thing

00:07:02.120 --> 00:07:07.600
as taking the reciprocal
of 2, which is 1/2

00:07:07.600 --> 00:07:13.271
and raising it to
the positive 4 power.