Squeeze theorem or sandwich theorem

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Kind: captions
Language: en

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We're now going
to think about one

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of my most favorite
theorems in mathematics,

00:00:05.230 --> 00:00:07.180
and that's the squeeze theorem.

00:00:07.180 --> 00:00:10.460
And one of the reasons that it's
one of my most favorite theorem

00:00:10.460 --> 00:00:13.865
in mathematics is that it
has the word "squeeze" in it,

00:00:13.865 --> 00:00:15.620
a word that you
don't see showing up

00:00:15.620 --> 00:00:17.170
in a lot of mathematics.

00:00:17.170 --> 00:00:18.820
But it is appropriately named.

00:00:18.820 --> 00:00:22.400
And this is oftentimes also
called the sandwich theorem,

00:00:22.400 --> 00:00:26.190
which is also an appropriate
name, as we'll see in a second.

00:00:26.190 --> 00:00:29.320
And since it can be called
the sandwich theorem,

00:00:29.320 --> 00:00:31.730
let's first just
think about an analogy

00:00:31.730 --> 00:00:35.340
to get the intuition behind
the squeeze or the sandwich

00:00:35.340 --> 00:00:36.390
theorem.

00:00:36.390 --> 00:00:38.830
Let's say that there
are three people.

00:00:38.830 --> 00:00:46.420
Let's say that there is
Imran, let's say there's Diya,

00:00:46.420 --> 00:00:49.160
and let's say there is Sal.

00:00:49.160 --> 00:00:52.370
And let's say that
Imran, on any given day,

00:00:52.370 --> 00:00:55.880
he always has the fewest
amount of calories.

00:00:55.880 --> 00:00:57.300
And Sal, on any
given day, always

00:00:57.300 --> 00:00:59.100
has the most number of calories.

00:00:59.100 --> 00:01:02.580
So in a given day,
we can always say

00:01:02.580 --> 00:01:15.190
Diya eats at least
as much as Imran.

00:01:15.190 --> 00:01:22.370
And then we can say Sal eats
at least as much-- that's

00:01:22.370 --> 00:01:26.070
just to repeat those
words-- as Diya.

00:01:26.070 --> 00:01:28.400
And so we could set up a
little inequality here.

00:01:28.400 --> 00:01:35.610
On a given day, we could
write that Imran's calories

00:01:35.610 --> 00:01:39.880
on a given day are going to be
less than or equal to Diya's

00:01:39.880 --> 00:01:47.390
calories on that same day,
which is going to be less than

00:01:47.390 --> 00:01:50.650
or equal to Sal's
calories on that same day.

00:01:55.340 --> 00:01:57.900
Now let's say that it's Tuesday.

00:01:57.900 --> 00:02:03.090
Let's say on
Tuesday you find out

00:02:03.090 --> 00:02:11.370
that Imran ate 1,500 calories.

00:02:11.370 --> 00:02:20.460
And on that same day, Sal
also ate 1,500 calories.

00:02:20.460 --> 00:02:22.860
So based on this,
how many calories

00:02:22.860 --> 00:02:25.650
must Diya have eaten that day?

00:02:25.650 --> 00:02:28.730
Well, she always eats at least
as many as Imran's, so she

00:02:28.730 --> 00:02:31.530
ate 1,500 calories or more.

00:02:31.530 --> 00:02:35.250
But she always has less than or
equal to the number of calories

00:02:35.250 --> 00:02:36.330
Sal eats.

00:02:36.330 --> 00:02:38.094
So it must be less
than or equal to 1,500.

00:02:38.094 --> 00:02:39.510
Well, there's only
one number that

00:02:39.510 --> 00:02:41.930
is greater than or equal
to 1,500 and less than

00:02:41.930 --> 00:02:45.960
or equal to 1,500, and
that is 1,500 calories.

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So Diya must have
eaten 1,500 calories.

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This is common sense.

00:02:50.530 --> 00:02:54.890
Diya must have had
1,500 calories.

00:02:54.890 --> 00:02:56.710
And the squeeze
theorem is essentially

00:02:56.710 --> 00:02:59.740
the mathematical version
of this for functions.

00:02:59.740 --> 00:03:04.170
And you could even view this is
Imran's calories as a function

00:03:04.170 --> 00:03:07.800
of the day, Sal's calories
as a function of the day,

00:03:07.800 --> 00:03:09.840
and Diya's calories as
a function of the day

00:03:09.840 --> 00:03:12.410
is always going to
be in between those.

00:03:12.410 --> 00:03:16.950
So now let's make this a
little bit more mathematical.

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So let me clear this out
so we can have some space

00:03:21.440 --> 00:03:23.430
to do some math in.

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So let's say that we
have the same analogy.

00:03:27.250 --> 00:03:29.620
So let's say that we
have three functions.

00:03:29.620 --> 00:03:32.650
Let's say f of x
over some interval

00:03:32.650 --> 00:03:35.900
is always less
than or equal to g

00:03:35.900 --> 00:03:40.340
of x over that same interval,
which is always less than

00:03:40.340 --> 00:03:45.530
or equal to h of x over
that same interval.

00:03:45.530 --> 00:03:47.305
So let me depict
this graphically.

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So that is my y-axis.

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This is my x-axis.

00:04:01.810 --> 00:04:03.940
And I'll just
depict some interval

00:04:03.940 --> 00:04:06.080
in the x-axis right over here.

00:04:06.080 --> 00:04:12.810
So let's say h of x looks
something like that.

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Let me make it more interesting.

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This is the x-axis.

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So let's say h of x looks
something like this.

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So that's my h of x.

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Let's say f of x looks
something like this.

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Maybe it does some interesting
things, and then it comes in,

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and then it goes up
like this, so f of x

00:04:35.260 --> 00:04:38.050
looks something like that.

00:04:38.050 --> 00:04:39.960
And then g of x,
for any x-value, g

00:04:39.960 --> 00:04:43.297
of x is always in
between these two.

00:04:43.297 --> 00:04:45.380
And I think you see where
the squeeze is happening

00:04:45.380 --> 00:04:48.780
and where the
sandwich is happening.

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If h of x and f of x were
bendy pieces of bread, g of x

00:04:52.300 --> 00:04:54.060
would be the meat of the bread.

00:04:54.060 --> 00:04:56.660
So it would look
something like this.

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Now, let's say that we know--
this is the analogous thing.

00:05:00.370 --> 00:05:03.290
On a particular day, Sal and
Imran ate the same amount.

00:05:03.290 --> 00:05:05.240
Let's say for a
particular x-value,

00:05:05.240 --> 00:05:09.990
the limit as f and h
approach that x-value, they

00:05:09.990 --> 00:05:12.000
approach is the same limit.

00:05:12.000 --> 00:05:14.310
So let's take this
x-value right over here.

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Let's say the x-value
is c right over there.

00:05:18.790 --> 00:05:23.070
And let's say that
the limit of f

00:05:23.070 --> 00:05:34.960
of x as x approaches c is
equal to L. And let's say

00:05:34.960 --> 00:05:45.105
that the limit as x approaches
c of h of x is also equal to L.

00:05:45.105 --> 00:05:48.010
So notice, as x
approaches c, h of x

00:05:48.010 --> 00:05:52.100
approaches L. As x approaches
c from either side, f of x

00:05:52.100 --> 00:05:53.560
approaches L.

00:05:53.560 --> 00:05:55.484
So these limits
have to be defined.

00:05:55.484 --> 00:05:56.900
Actually, the
functions don't have

00:05:56.900 --> 00:05:58.610
to be defined at x approaches c.

00:05:58.610 --> 00:06:00.240
Just over this
interval, they have

00:06:00.240 --> 00:06:02.120
to be defined as we approach it.

00:06:02.120 --> 00:06:04.530
But over this interval,
this has to be true.

00:06:04.530 --> 00:06:06.805
And if these limits right
over here are defined

00:06:06.805 --> 00:06:09.350
and because we know that g
of x is always sandwiched in

00:06:09.350 --> 00:06:11.830
between these two
functions, therefore,

00:06:11.830 --> 00:06:14.530
on that day or
for that x-value--

00:06:14.530 --> 00:06:16.800
I should get out of that
food-eating analogy--

00:06:16.800 --> 00:06:21.910
this tells us if all of this
is true over this interval,

00:06:21.910 --> 00:06:31.560
this tells us that the limit
as x approaches c of g of x

00:06:31.560 --> 00:06:34.050
must also be equal to L.

00:06:34.050 --> 00:06:36.280
And once again, this
is common sense.

00:06:36.280 --> 00:06:39.750
f of x is approaching
L, h of x is approaching

00:06:39.750 --> 00:06:42.130
L, g of x is sandwiched
in between it.

00:06:42.130 --> 00:06:47.444
So it also has to
be approaching L.

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And you might say, well,
this is common sense.

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Why is this useful?

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Well, as you'll see, this
is useful for finding

00:06:53.640 --> 00:06:55.830
the limits of some
wacky functions.

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If you can find a function
that's always greater than it

00:06:58.350 --> 00:07:00.145
and a function that's
always less than it,

00:07:00.145 --> 00:07:02.270
and you can find the limit
as they approach some c,

00:07:02.270 --> 00:07:04.650
and it's the same
limit, then you

00:07:04.650 --> 00:07:06.490
know that that wacky
function in between

00:07:06.490 --> 00:07:09.997
is going to approach
that same limit.


Dividing fractions example

WEBVTT
Kind: captions
Language: en

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So let's calculate
what 2/5 divided by 7/3

00:00:04.900 --> 00:00:06.880
is, and I encourage
you to pause this video

00:00:06.880 --> 00:00:09.340
and try to calculate
this on your own.

00:00:09.340 --> 00:00:11.720
Well we just have to
remind ourselves that this

00:00:11.720 --> 00:00:21.130
is going to be the exact
same thing as 2 over 5 times

00:00:21.130 --> 00:00:26.120
the reciprocal of 7/3,
which is 3 over 7.

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And then multiplying
two fractions

00:00:30.830 --> 00:00:32.200
is pretty straightforward.

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This is just going to
be equal to the product

00:00:34.200 --> 00:00:35.520
of the numerators.

00:00:35.520 --> 00:00:44.710
So 2 times 3 over the
product of the denominators,

00:00:44.710 --> 00:00:54.380
over 5 times 7-- I'm trying to
keep the colors consistent--

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which of course is going
to be equal to 2 times 3

00:00:59.780 --> 00:01:02.210
is equal to 6.

00:01:02.210 --> 00:01:05.590
And 5 times 7-- I'll do this
in a new color, let's see,

00:01:05.590 --> 00:01:10.320
I haven't used this shade of
blue yet-- 5 times 7 of course

00:01:10.320 --> 00:01:12.250
is equal to 35.

00:01:12.250 --> 00:01:14.966
So this is equal to 6/35.


Multiplying challenging decimals

WEBVTT
Kind: captions
Language: en

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Let's multiply 1.21,
or 1 and 21 hundredths,

00:00:04.660 --> 00:00:09.336
times 43 thousandths, or 0.043.

00:00:09.336 --> 00:00:10.960
And I encourage you
to pause this video

00:00:10.960 --> 00:00:13.540
and try it on your own.

00:00:13.540 --> 00:00:16.490
So let's just think about a
very similar problem but one

00:00:16.490 --> 00:00:18.560
where essentially we
don't write the decimals.

00:00:18.560 --> 00:00:23.550
Let's just think about
multiplying 121 times 43,

00:00:23.550 --> 00:00:25.517
which we know how to do.

00:00:25.517 --> 00:00:27.100
So let's just think
about this problem

00:00:27.100 --> 00:00:29.269
first as kind of
a simplification,

00:00:29.269 --> 00:00:31.560
and then we'll think about
how to get from this product

00:00:31.560 --> 00:00:33.120
to this product.

00:00:33.120 --> 00:00:38.020
So we can start with-- so we're
going to say 3 times 1 is 3.

00:00:38.020 --> 00:00:39.920
3 times 2 is 6.

00:00:39.920 --> 00:00:42.650
3 times 1 is 3.

00:00:42.650 --> 00:00:45.927
3 times 121 is 363.

00:00:45.927 --> 00:00:47.760
And now we're going to
go to the tens place,

00:00:47.760 --> 00:00:49.950
so this is a 40 right over here.

00:00:49.950 --> 00:00:52.750
So since we're in the tens
place, let's put a 0 there.

00:00:52.750 --> 00:00:56.900
40 times 1 is 40.

00:00:56.900 --> 00:01:00.682
40 times 20 is 800.

00:01:00.682 --> 00:01:04.970
40 times 100 is 4,000.

00:01:04.970 --> 00:01:07.410
And we've already known
how to do this in the past,

00:01:07.410 --> 00:01:10.210
and now we can just add
all of this together.

00:01:10.210 --> 00:01:14.880
And we get-- let me do a new
color here-- 3 plus 0 is 3.

00:01:14.880 --> 00:01:17.940
6 plus 4 is 10.

00:01:17.940 --> 00:01:21.340
1 plus 3 plus 8 is 12.

00:01:21.340 --> 00:01:23.660
1 plus 4 is 5.

00:01:23.660 --> 00:01:27.970
So 121 times 43 is 5,203.

00:01:27.970 --> 00:01:31.240
Now, how is this useful for
figuring out this product?

00:01:31.240 --> 00:01:36.890
Well, to go from
1.21 to 121, we're

00:01:36.890 --> 00:01:40.020
essentially multiplying by 100.

00:01:40.020 --> 00:01:40.610
Right?

00:01:40.610 --> 00:01:44.600
We're moving the decimal two
places over to the right.

00:01:44.600 --> 00:01:49.930
And to go from 0.043 to
43, what are we doing?

00:01:49.930 --> 00:01:51.700
We're removing the
decimal, so we're

00:01:51.700 --> 00:01:55.630
multiplying by ten,
hundred, thousand.

00:01:55.630 --> 00:01:59.730
We're multiplying by 1,000.

00:01:59.730 --> 00:02:02.480
So to go from this
product to this product

00:02:02.480 --> 00:02:05.720
or to this product, we
essentially multiplied by 100,

00:02:05.720 --> 00:02:08.530
and we multiplied by 1,000.

00:02:08.530 --> 00:02:14.350
So then to go back to this
product, we have to divide.

00:02:14.350 --> 00:02:20.700
We should divide by
100 and then divide

00:02:20.700 --> 00:02:25.650
by 1,000, which is equivalent
to dividing by 100,000.

00:02:25.650 --> 00:02:26.920
But let's do that.

00:02:26.920 --> 00:02:32.569
So let's rewrite this
number here, so 5,203.

00:02:32.569 --> 00:02:33.985
Actually let me
write it like this

00:02:33.985 --> 00:02:39.570
just so it's a little
bit more aligned, 5,203.

00:02:39.570 --> 00:02:42.240
And we could imagine a
decimal point right over here.

00:02:42.240 --> 00:02:47.150
If we divide by 100-- so you
divide by 10, divide by 100--

00:02:47.150 --> 00:02:49.520
and then we want to
divide by another 1,000.

00:02:49.520 --> 00:02:55.620
So divide by 10, divide
by 100, divide by 1,000.

00:02:55.620 --> 00:02:59.510
So our decimal point is
going to go right over there,

00:02:59.510 --> 00:03:00.260
and we're done.

00:03:00.260 --> 00:03:06.900
1.21 times 0.043 is 0.05203.

00:03:06.900 --> 00:03:10.270
So one way you could think about
it is just multiply these two

00:03:10.270 --> 00:03:13.240
numbers as if there
were no decimals there.

00:03:13.240 --> 00:03:15.394
Then you could count
how many digits

00:03:15.394 --> 00:03:17.060
are to the right of
the decimal, and you

00:03:17.060 --> 00:03:24.500
see that there are one,
two, three, four, five

00:03:24.500 --> 00:03:26.980
digits to the right
of the decimal,

00:03:26.980 --> 00:03:30.790
and so in your product, you
should have one, two, three,

00:03:30.790 --> 00:03:35.680
four, five digits to the
right of the decimal.

00:03:35.680 --> 00:03:37.497
Why is that the case?

00:03:37.497 --> 00:03:39.580
Well, when you ignored the
decimals, when you just

00:03:39.580 --> 00:03:43.360
pretended that this
was 121 times 43,

00:03:43.360 --> 00:03:46.570
you essentially multiplied
this times 100,000--

00:03:46.570 --> 00:03:49.985
by 100 and 1,000-- and so to
get from the product you get

00:03:49.985 --> 00:03:51.860
without the decimals to
the one that you need

00:03:51.860 --> 00:03:55.450
with the decimals, you have to
then divide by 100,000 again.

00:03:55.450 --> 00:03:57.520
Multiplied by 100,000
is essentially

00:03:57.520 --> 00:04:02.210
equivalent to moving the decimal
place five places to the right,

00:04:02.210 --> 00:04:06.110
and then dividing by 100,000
is equivalent to the moving

00:04:06.110 --> 00:04:08.540
the decimal five
digits to the left.

00:04:08.540 --> 00:04:14.530
So divide by 10, divide
by 100, divide by 1,000,

00:04:14.530 --> 00:04:18.475
divide by 10,000,
divide by 100,000.

00:04:18.475 --> 00:04:20.779
And either way, we are done.

00:04:20.779 --> 00:04:24.660
This is what we get.


Multiplying decimals: place value and reordering

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:05.230
Let's see if we can
calculate 2.91 times 3.2.

00:00:05.230 --> 00:00:07.020
And I encourage you
to pause this video

00:00:07.020 --> 00:00:09.180
and try it out on your own.

00:00:09.180 --> 00:00:15.030
So the way I'm going to
think about it is 2.91

00:00:15.030 --> 00:00:22.655
is the same thing as
291 divided by 10.

00:00:22.655 --> 00:00:25.620
Or not divided by
10, divided by 100.

00:00:25.620 --> 00:00:27.660
And we know that if you
divide something by 100,

00:00:27.660 --> 00:00:29.451
you are going to move
the decimal place two

00:00:29.451 --> 00:00:31.770
places to the left-- one, two.

00:00:31.770 --> 00:00:34.310
And you would end up at 2.91.

00:00:34.310 --> 00:00:38.720
It also make sense, if I take
2, and I multiply it by 100,

00:00:38.720 --> 00:00:39.830
I'd get 200.

00:00:39.830 --> 00:00:43.970
Or if I take 200 and divided
by 100, I would get 2.

00:00:43.970 --> 00:00:47.140
So it makes sense that
2.91 is the same thing

00:00:47.140 --> 00:00:50.210
as 291 divided by 100.

00:00:50.210 --> 00:00:55.780
Similarlarly-- I can
never say that word-- 3.2

00:00:55.780 --> 00:00:57.050
can be rewritten.

00:00:57.050 --> 00:01:03.850
It's the same thing
as 32 divided by 10.

00:01:03.850 --> 00:01:06.120
Now, why is all of
this interesting?

00:01:06.120 --> 00:01:20.020
Well, I could rewrite 2.91 times
3.2 as being the same thing as.

00:01:20.020 --> 00:01:30.290
Instead of 2.91, I can
write 291 divided by 100.

00:01:30.290 --> 00:01:34.560
And then times--
instead of writing 3.2,

00:01:34.560 --> 00:01:39.510
I could write 32 divided by 10.

00:01:39.510 --> 00:01:42.480
And this can be
rewritten as-- this

00:01:42.480 --> 00:01:55.310
is going to be equal to 291
times 32 divided by 100.

00:01:55.310 --> 00:02:04.700
I'm just reordering this--
divided by 100, divided by 10.

00:02:04.700 --> 00:02:08.622
Or I could rewrite this.

00:02:08.622 --> 00:02:12.316
This is equal to 291 times 32.

00:02:15.090 --> 00:02:18.320
If I divide by 100 and
then I divide by 10 again,

00:02:18.320 --> 00:02:21.076
I'm essentially
dividing by 1,000.

00:02:21.076 --> 00:02:23.610
So this part right
over here, I could

00:02:23.610 --> 00:02:26.750
rewrite as dividing by 1,000.

00:02:26.750 --> 00:02:28.260
Now, why is this interesting?

00:02:28.260 --> 00:02:31.570
Well, I already know how
to multiply 291 times 32.

00:02:31.570 --> 00:02:33.340
And then we know how
to move the decimal

00:02:33.340 --> 00:02:35.140
so that when we divide by 1,000.

00:02:35.140 --> 00:02:38.200
So let's calculate 291 times 32.

00:02:38.200 --> 00:02:39.990
Let me write it right over here.

00:02:39.990 --> 00:02:44.498
291 times 32.

00:02:47.370 --> 00:02:49.690
Notice, I've just
essentially rewritten this

00:02:49.690 --> 00:02:50.940
without the decimals.

00:02:50.940 --> 00:02:52.797
So this right over
here-- but of course,

00:02:52.797 --> 00:02:54.630
these are different
quantities than this one

00:02:54.630 --> 00:02:55.421
is right over here.

00:02:55.421 --> 00:02:57.250
To go from this product
to this product,

00:02:57.250 --> 00:02:58.960
I have to divide by 1,000.

00:02:58.960 --> 00:03:00.637
But let's just think about this.

00:03:00.637 --> 00:03:02.720
We already know how to
compute this type of thing.

00:03:02.720 --> 00:03:04.870
2 times 1 is 2.

00:03:04.870 --> 00:03:07.810
2 times 9 is 18.

00:03:07.810 --> 00:03:08.500
Carry the 1.

00:03:08.500 --> 00:03:12.900
2 times 2 is 4, plus 1 is 5.

00:03:12.900 --> 00:03:15.240
And now we can
think about the 3.

00:03:15.240 --> 00:03:18.280
3 times 1-- oh, let
me throw a 0 here.

00:03:18.280 --> 00:03:20.100
Because this isn't a 3.

00:03:20.100 --> 00:03:21.210
This is now a 30.

00:03:21.210 --> 00:03:22.790
So this is in the tens place.

00:03:22.790 --> 00:03:24.295
So that's why I put a 0 there.

00:03:24.295 --> 00:03:28.800
So 30 times 1 is 30.

00:03:28.800 --> 00:03:31.610
That's why we say 3
times 1 is 3, but notice,

00:03:31.610 --> 00:03:33.430
it's in the tens
place right now.

00:03:33.430 --> 00:03:38.410
And then 3 times 9 is 27.

00:03:38.410 --> 00:03:40.294
Carry the 2.

00:03:40.294 --> 00:03:45.980
3 times 2 is 6, plus 2 is 8.

00:03:45.980 --> 00:03:49.185
And now we can add.

00:03:49.185 --> 00:03:51.250
And we would get 2.

00:03:51.250 --> 00:03:53.810
8 plus 3 is 11.

00:03:53.810 --> 00:03:56.471
6 plus 3 is 13.

00:03:56.471 --> 00:03:57.860
And then you get 9.

00:03:57.860 --> 00:04:00.070
So you get 9,312.

00:04:00.070 --> 00:04:09.626
So this is going to be equal
to 9,312 divided by 1,000.

00:04:09.626 --> 00:04:12.390
And what's this
going to be equal to?

00:04:12.390 --> 00:04:16.180
Well, if we start
with 9,312-- and let

00:04:16.180 --> 00:04:17.790
me throw a decimal there.

00:04:17.790 --> 00:04:20.450
Dividing by 1,000 is
equivalent to moving

00:04:20.450 --> 00:04:23.130
the decimal over three
places to the left.

00:04:23.130 --> 00:04:27.660
So you divide by 10, divide
by 100, divide by 1,000.

00:04:27.660 --> 00:04:31.085
So that's going to be 9.312.

00:04:31.085 --> 00:04:44.247
So if you divide by 1,000,
you will get to 9.312.

00:04:44.247 --> 00:04:45.705
Let me write the
decimal in purple.

00:04:51.320 --> 00:04:54.550
Now, there's something
very interesting here.

00:04:54.550 --> 00:04:57.550
In our original, when
we wrote the expression,

00:04:57.550 --> 00:05:04.780
we had one, two, three total
numbers behind the decimal.

00:05:04.780 --> 00:05:08.180
And then over here,
we have one, two,

00:05:08.180 --> 00:05:12.330
three total numbers to
the right of the decimal.

00:05:12.330 --> 00:05:13.850
Why is this?

00:05:13.850 --> 00:05:17.330
Well, let's think about it.

00:05:17.330 --> 00:05:21.410
We re-expressed this
as 291 divided by 100.

00:05:21.410 --> 00:05:25.662
And this is 32 divided by 10.

00:05:25.662 --> 00:05:28.110
Dividing by 100 and
dividing by 10--

00:05:28.110 --> 00:05:31.740
this essentially accounts for
these three decimal places.

00:05:31.740 --> 00:05:34.300
So we essentially get rid
of those decimal places.

00:05:34.300 --> 00:05:37.210
But then we have to reintroduce
those three decimal places

00:05:37.210 --> 00:05:38.490
by dividing.

00:05:38.490 --> 00:05:42.150
Or we have to shift-- here we
shift the decimal an aggregate

00:05:42.150 --> 00:05:46.520
to the right three times--
one, two, and then three.

00:05:46.520 --> 00:05:48.820
Now, in order to make sure
we get the right product,

00:05:48.820 --> 00:05:51.190
we've got to shift
it back to the left.

00:05:51.190 --> 00:05:53.830
So we're shifting
it one, two, three.

00:05:53.830 --> 00:05:55.800
So we went from this to this.

00:05:55.800 --> 00:05:58.156
For the whole product,
it was like multiplying.

00:06:01.240 --> 00:06:06.860
We essentially, to go from here
to here, we multiplied by 100.

00:06:06.860 --> 00:06:09.870
To go from here to here,
we multiplied by 10.

00:06:09.870 --> 00:06:12.900
So in aggregate, we
multiplied by 1,000,

00:06:12.900 --> 00:06:14.730
if you think about
both of these.

00:06:14.730 --> 00:06:18.080
And so now we have to divide by
1,000 to get the right value.

00:06:18.080 --> 00:06:24.100
So that's why three spaces
to the right of the decimal

00:06:24.100 --> 00:06:27.527
here, three digits to the
right of the decimal here.


Introduction to multiplying decimals

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:07.582
Let's see if we can
multiply 9 times 0.6.

00:00:07.582 --> 00:00:14.052
Or another way to write it, we
want to calculate 9 times 0.6.

00:00:14.052 --> 00:00:16.260
I'll write it like this-- 0.6.

00:00:16.260 --> 00:00:20.120
We want to figure out
what this is equal to.

00:00:20.120 --> 00:00:21.704
And I encourage you
to pause the video

00:00:21.704 --> 00:00:23.245
and try to figure
it out on your own.

00:00:23.245 --> 00:00:25.204
And I'll give you a
little bit of a hint.

00:00:25.204 --> 00:00:34.740
0.6 is the same thing
as 6 divided by 10.

00:00:34.740 --> 00:00:38.880
We know that if we start with
6, which we could write as 6.0,

00:00:38.880 --> 00:00:41.930
and if you were to divide
it by 10, dividing by 10

00:00:41.930 --> 00:00:44.880
is equivalent to moving
the decimal place one

00:00:44.880 --> 00:00:46.750
place to the left.

00:00:46.750 --> 00:00:50.220
So 6 divided by 10 is 0.6.

00:00:50.220 --> 00:00:54.730
We are moving the decimal
one place to the left.

00:00:54.730 --> 00:00:56.890
So I'm assuming you
given a go at it.

00:00:56.890 --> 00:00:59.420
But what I'm going
to do is use this

00:00:59.420 --> 00:01:02.540
that we already know to rewrite
what we're trying to multiply.

00:01:02.540 --> 00:01:07.650
So 9 times 0.6 is the
same thing is 9 times--

00:01:07.650 --> 00:01:14.350
0.6 is 6 divided by 10.

00:01:14.350 --> 00:01:16.270
And this expression
right over here,

00:01:16.270 --> 00:01:19.430
we could either do the 6 divided
by 10 first, in which case

00:01:19.430 --> 00:01:23.790
we would get 0.6, and this
would turn into this problem.

00:01:23.790 --> 00:01:28.030
Or we could do the
9 times 6 first.

00:01:28.030 --> 00:01:31.670
And so let's do 9 times 6,
which we know how to calculate,

00:01:31.670 --> 00:01:33.901
and then divide by 10, which
we also know how to do.

00:01:33.901 --> 00:01:34.400
That.

00:01:34.400 --> 00:01:36.620
all about just moving
the decimal place.

00:01:36.620 --> 00:01:42.130
So we could write 9 times 6.

00:01:42.130 --> 00:01:45.218
9 times 6, we
already know, is 54.

00:01:45.218 --> 00:01:48.760
I'll do that in orange--
is going to be 54.

00:01:48.760 --> 00:01:51.480
So this right over here is 54.

00:01:51.480 --> 00:01:53.610
And now to get to
this expression,

00:01:53.610 --> 00:01:55.310
we have to divide by 10.

00:01:55.310 --> 00:01:58.295
We have to divide by 10.

00:01:58.295 --> 00:02:00.830
And what happens when we
divide something by 10?

00:02:00.830 --> 00:02:03.520
And we've seen this in previous
videos, why this is the case.

00:02:03.520 --> 00:02:06.330
This is all about what
decimal notation means.

00:02:06.330 --> 00:02:10.889
Each place represents 10
times as much as the place

00:02:10.889 --> 00:02:14.310
to its right, or each place
represents 1/10 of the place

00:02:14.310 --> 00:02:14.855
to the left.

00:02:14.855 --> 00:02:18.290
So 54 divided by
10, this is going

00:02:18.290 --> 00:02:22.154
to be-- you could start with 54.

00:02:22.154 --> 00:02:25.250
And I'll put a 0 here
after the decimal.

00:02:25.250 --> 00:02:27.050
And when you divide
by 10, that's

00:02:27.050 --> 00:02:29.720
equivalent of shifting the
decimal one to the left.

00:02:29.720 --> 00:02:32.091
This is going to
be equal to 5.4.

00:02:32.091 --> 00:02:33.760
And that should
make sense to you.

00:02:33.760 --> 00:02:36.310
5 times 10 is 50.

00:02:36.310 --> 00:02:37.950
0.4 times 10 is 4.

00:02:37.950 --> 00:02:42.924
So it makes sense that
54 divided by 10--

00:02:42.924 --> 00:02:43.840
I shouldn't say equal.

00:02:43.840 --> 00:02:50.705
I'd write 54 divided
by 10 is equal to 5.4.

00:02:50.705 --> 00:02:54.940
So this right over
here is equal to 5.4,

00:02:54.940 --> 00:02:56.820
and that's what this is.

00:02:56.820 --> 00:03:00.660
This is equal to 5.4.

00:03:00.660 --> 00:03:03.910
Notice, 9 times 6 is 54.

00:03:03.910 --> 00:03:08.250
9 times 0.6 is 5.4.

00:03:08.250 --> 00:03:10.500
Now you might see a
little pattern here.

00:03:10.500 --> 00:03:14.690
Between these two numbers,
I had exactly one number

00:03:14.690 --> 00:03:17.000
to the right of the decimal.

00:03:17.000 --> 00:03:19.870
When I take its product, let's
say I ignored the decimal.

00:03:19.870 --> 00:03:22.290
I just said 9 times 6,
I would've gotten 54.

00:03:22.290 --> 00:03:25.570
But then I have to
divide by 10 in order

00:03:25.570 --> 00:03:27.270
to take account of
the decimal, take

00:03:27.270 --> 00:03:29.010
account of the fact
this wasn't a 6.

00:03:29.010 --> 00:03:30.152
This was a 6/10.

00:03:30.152 --> 00:03:32.989
And so I have one number to
the right of the decimal here.

00:03:32.989 --> 00:03:34.530
And I want to you
to think about that

00:03:34.530 --> 00:03:36.160
whether that's a
general principle.

00:03:36.160 --> 00:03:39.300
Can we just count the
total numbers of digits

00:03:39.300 --> 00:03:41.210
to the right of the
decimals and then

00:03:41.210 --> 00:03:44.800
our product is going to have
the same number of digits

00:03:44.800 --> 00:03:46.390
to right of the decimal?

00:03:46.390 --> 00:03:49.210
I'll let you to
think about that.


Capstone exponent properties example

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.190
So we have this equation
right over here.

00:00:03.190 --> 00:00:07.380
12 to the negative 2
power, times 1 over 72

00:00:07.380 --> 00:00:12.520
to the fifth power, is equal
to 2 to the a power times 3

00:00:12.520 --> 00:00:14.440
to the b power.

00:00:14.440 --> 00:00:17.760
And what I want you to do
now is pause this video

00:00:17.760 --> 00:00:22.850
and find values of a and b that
satisfy this equation, where

00:00:22.850 --> 00:00:25.520
a and b are two integers.

00:00:25.520 --> 00:00:28.360
And I encourage you to use
all of the exponent properties

00:00:28.360 --> 00:00:32.780
that we have now learned
in order to tackle this.

00:00:32.780 --> 00:00:34.810
So I'm assuming you've
given a go at it,

00:00:34.810 --> 00:00:37.360
so now we can work
through this together.

00:00:37.360 --> 00:00:41.130
So what I'm going to try to do,
is get these numbers, like 12,

00:00:41.130 --> 00:00:43.640
and these numbers
like 72, I want

00:00:43.640 --> 00:00:49.860
to express them as products
of powers of 2 and powers of 3

00:00:49.860 --> 00:00:55.930
So 12, for example, is
equivalent to 3 times 4.

00:00:55.930 --> 00:00:59.980
So I could write
this as 3 times 4.

00:00:59.980 --> 00:01:04.050
And all of that is going to be
raised to the negative 2 power.

00:01:04.050 --> 00:01:05.670
Now, why am I
interested in 3 times 4?

00:01:05.670 --> 00:01:10.150
Well, 3 is clearly a power of 3,
and 4 is clearly a power of 2.

00:01:10.150 --> 00:01:11.610
This is 2 squared,
we're actually

00:01:11.610 --> 00:01:14.120
going to write it as
2 squared in a second.

00:01:14.120 --> 00:01:17.090
So this is 3 times 4 to
the negative 2 power.

00:01:17.090 --> 00:01:21.750
And then 1 over 72
could be rewritten

00:01:21.750 --> 00:01:26.850
as-- so it's going to be
times-- 1 over 8 times 9,

00:01:26.850 --> 00:01:29.110
72 is 8 times 9.

00:01:29.110 --> 00:01:34.220
8 is a power of 2,
9 is a power of 3

00:01:34.220 --> 00:01:38.230
And, of course, we're going to
raise that, to the fifth power.

00:01:38.230 --> 00:01:40.950
And that's going to be equal to
2 to the a, times 3 to the b.

00:01:40.950 --> 00:01:45.350
I'm just going to keep trying
to simplify the left hand,

00:01:45.350 --> 00:01:47.710
in terms of powers
of 2 and 3, and then

00:01:47.710 --> 00:01:52.510
see if we can match what a and
b need to actually be equal to.

00:01:52.510 --> 00:01:54.800
So let's look at
this right over here.

00:01:54.800 --> 00:01:57.180
So this, if we have the
product of two numbers raised

00:01:57.180 --> 00:02:00.590
to an exponent, we know
from our exponent properties

00:02:00.590 --> 00:02:03.930
that this is going to be
each of those numbers, raised

00:02:03.930 --> 00:02:05.820
to the exponent,
the product of that.

00:02:05.820 --> 00:02:08.479
Or the product of each of those
numbers raised to the exponent.

00:02:08.479 --> 00:02:12.230
So it's going to be 3
to the negative 2 times

00:02:12.230 --> 00:02:14.570
4 to the negative 2 power.

00:02:17.930 --> 00:02:19.596
And now this right
over here, before,

00:02:19.596 --> 00:02:20.970
let me just rewrite
it like this,

00:02:20.970 --> 00:02:31.100
this is going to be
the same thing, times 1

00:02:31.100 --> 00:02:35.115
over 8 times 1 over 9.

00:02:35.115 --> 00:02:39.620
And we're going to raise
that to the fifth power.

00:02:39.620 --> 00:02:42.240
Now 3 to the negative 2,
we could just keep as 3

00:02:42.240 --> 00:02:43.009
to the negative 2.

00:02:43.009 --> 00:02:44.800
That seems like a nice
form to leave it in.

00:02:44.800 --> 00:02:48.460
So let me write that as 3
to the negative 2 power.

00:02:48.460 --> 00:02:51.440
Now what is 4 to the
negative 2 power?

00:02:51.440 --> 00:02:56.850
Well, 4 is the same
thing as 2 squared.

00:02:56.850 --> 00:03:00.896
So it's going to be times
2 squared to the negative 2

00:03:00.896 --> 00:03:01.395
power.

00:03:04.510 --> 00:03:06.590
Now what is 1 over 8?

00:03:06.590 --> 00:03:09.300
What is 1/8?

00:03:09.300 --> 00:03:13.160
You could view this as 8
to the negative 1 power.

00:03:13.160 --> 00:03:16.050
Or you could view
this as 1 over 2

00:03:16.050 --> 00:03:19.790
to the third power, which
is the same thing as 2

00:03:19.790 --> 00:03:21.449
to the negative 3 power.

00:03:21.449 --> 00:03:23.240
And these are consistent,
because remember,

00:03:23.240 --> 00:03:25.640
8 is the same thing
as 2 to the third.

00:03:25.640 --> 00:03:27.730
So this is 2 to the
third to the negative 1

00:03:27.730 --> 00:03:30.430
power, which is 2 to the
3 times negative 1, which

00:03:30.430 --> 00:03:32.650
is 2 to the negative 3 power.

00:03:32.650 --> 00:03:35.640
So 1/8 is 2 to the
negative third power.

00:03:35.640 --> 00:03:42.430
And by that same logic,
1/9 is equal to 9

00:03:42.430 --> 00:03:44.150
to the negative 1 power.

00:03:44.150 --> 00:03:48.850
Or you could view it as 1 over
3 squared, which is equal to 3

00:03:48.850 --> 00:03:50.350
to the negative 2.

00:03:50.350 --> 00:03:52.510
And these, once again,
are consistent because 9

00:03:52.510 --> 00:03:55.840
is 3 squared, and you're raising
it to the negative 1 power.

00:03:55.840 --> 00:03:58.530
So, once again, that's 3
to the 2 times negative 1,

00:03:58.530 --> 00:04:00.570
3 to the negative 2 power.

00:04:00.570 --> 00:04:04.290
So this is going to be equal
to-- we're raising everything

00:04:04.290 --> 00:04:06.460
to the fifth power.

00:04:06.460 --> 00:04:10.940
So 1/8, we just established,
is the same thing.

00:04:10.940 --> 00:04:15.480
1/8 we just-- having trouble
changing colors-- established

00:04:15.480 --> 00:04:18.970
is 2 to the negative 3 power.

00:04:18.970 --> 00:04:24.310
And 1/9-- I'm running
out of colors--

00:04:24.310 --> 00:04:31.060
is 3 to the negative 2 power.

00:04:31.060 --> 00:04:32.470
So let's keep simplifying this.

00:04:32.470 --> 00:04:39.840
We have our 3 to the
negative 2 power.

00:04:39.840 --> 00:04:42.580
Now this part
right over here-- I

00:04:42.580 --> 00:04:45.360
could introduce purple
again-- that part,

00:04:45.360 --> 00:04:47.520
that's 2 to the
second power, and then

00:04:47.520 --> 00:04:49.020
that raised to the
negative 2 power.

00:04:49.020 --> 00:04:52.682
Well, we've already seen, that's
the same thing as taking 2

00:04:52.682 --> 00:04:55.090
to the product of
these two exponents.

00:04:55.090 --> 00:04:59.930
2 times negative 2
is negative 4 power.

00:04:59.930 --> 00:05:02.400
And then, we're going
to take each of these,

00:05:02.400 --> 00:05:04.700
we've already seen, taking
the product of two numbers

00:05:04.700 --> 00:05:06.400
and raising it to
an exponent, that's

00:05:06.400 --> 00:05:08.126
the same thing as
taking each of those,

00:05:08.126 --> 00:05:10.500
raising it to the exponent,
or taking the product of each

00:05:10.500 --> 00:05:12.290
of those, raised
to the exponent.

00:05:12.290 --> 00:05:16.430
So it's going to be
times 2 to the negative 3

00:05:16.430 --> 00:05:19.480
raised to the fifth power.

00:05:19.480 --> 00:05:23.510
Times, we're in the
home stretch, times 3

00:05:23.510 --> 00:05:28.410
to the negative 2 raised
to the fifth power.

00:05:28.410 --> 00:05:33.230
So once again, this is going to
be 3 to the negative 2 times 2

00:05:33.230 --> 00:05:35.830
to the negative 4, I
think you see the end now.

00:05:35.830 --> 00:05:36.800
And now what is this?

00:05:36.800 --> 00:05:38.604
2 to the negative 3
to the fifth, and then

00:05:38.604 --> 00:05:40.020
that raised to the
fifth, well, we

00:05:40.020 --> 00:05:41.230
multiply these two exponents.

00:05:41.230 --> 00:05:44.340
That's going to be 2 to
the negative 15, times 2

00:05:44.340 --> 00:05:48.850
to the negative 15 power, and
then what is this going to be?

00:05:48.850 --> 00:05:51.510
Negative 2 times
5 is negative 10.

00:05:51.510 --> 00:05:57.880
So this is going to be 3
to the negative 10th power.

00:05:57.880 --> 00:06:00.210
And so, what's this
going to be equal to?

00:06:00.210 --> 00:06:04.500
Well if we multiply this one--
let me introduce a new color--

00:06:04.500 --> 00:06:08.820
if we multiply 3 to the negative
2, times 3 to the negative 10,

00:06:08.820 --> 00:06:11.640
we have the same base, we
would add the exponents.

00:06:11.640 --> 00:06:14.930
So that's going to be 3 to the
negative 2 plus negative 10,

00:06:14.930 --> 00:06:17.300
which is the
negative 12th power.

00:06:17.300 --> 00:06:20.330
And then we have a 2
to the negative 4 times

00:06:20.330 --> 00:06:21.710
2 to the negative 15.

00:06:21.710 --> 00:06:25.715
That's going to be 2 to
the negative 19th power.

00:06:29.080 --> 00:06:31.466
And we want that to be
equal to 2 to the a.

00:06:31.466 --> 00:06:33.840
Now, we go to the beginning,
moving back to our equation,

00:06:33.840 --> 00:06:46.280
we want that to be equal to
2 to the a times 3 to the b.

00:06:46.280 --> 00:06:48.440
Well, we have 2 to the
negative 19th power,

00:06:48.440 --> 00:06:53.220
so let's let a equal
to negative 19.

00:06:53.220 --> 00:06:55.850
And we have 3 to
the negative 12,

00:06:55.850 --> 00:07:01.050
so let's set b is
equal to negative 12.

00:07:01.050 --> 00:07:02.960
And we are done.

00:07:02.960 --> 00:07:08.280
So all we did, this was just
an aggressive application

00:07:08.280 --> 00:07:11.154
of pretty much all of
the exponent properties

00:07:11.154 --> 00:07:12.820
that we actually know,
and it's actually

00:07:12.820 --> 00:07:15.476
kind of a satisfying problem.


Products and exponents raised to an exponent properties

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.670
And now I want to go over some
of the other core exponent

00:00:03.670 --> 00:00:04.170
properties.

00:00:04.170 --> 00:00:06.340
But they really just fall
out of what we already

00:00:06.340 --> 00:00:08.010
know about exponents.

00:00:08.010 --> 00:00:11.960
Let's say I have two
numbers, a and b.

00:00:11.960 --> 00:00:14.980
And I'm going to raise it to--
I could do it in the abstract.

00:00:14.980 --> 00:00:17.451
I could raise it to the c power.

00:00:17.451 --> 00:00:19.200
But I'll do it a little
bit more concrete.

00:00:19.200 --> 00:00:22.310
Let's raise it to
the fourth power.

00:00:22.310 --> 00:00:24.410
What is that going
to be equal to?

00:00:24.410 --> 00:00:27.450
Well that's going
to be equal to-- I

00:00:27.450 --> 00:00:29.110
could write it like this.

00:00:29.110 --> 00:00:31.130
Copy and paste this,
copy and paste.

00:00:31.130 --> 00:00:40.290
That's going to be equal to
ab times ab times ab times ab

00:00:40.290 --> 00:00:43.650
times ab.

00:00:43.650 --> 00:00:45.600
But what is that equal to?

00:00:45.600 --> 00:00:48.200
Well when you just multiply
a bunch of numbers like this

00:00:48.200 --> 00:00:51.660
it doesn't matter what order
you're going to multiply it in.

00:00:51.660 --> 00:00:55.060
This right over here is going
to be equivalent to a times

00:00:55.060 --> 00:00:59.329
a times a times
a times-- We have

00:00:59.329 --> 00:01:01.370
four b's as well that
we're multiplying together.

00:01:01.370 --> 00:01:05.990
Times b times b times b times b.

00:01:05.990 --> 00:01:07.590
And what is that equal to?

00:01:07.590 --> 00:01:13.480
Well this right over here
is a to the fourth power.

00:01:13.480 --> 00:01:21.390
And this right over here
is b to the fourth power.

00:01:21.390 --> 00:01:23.860
And so you see, if you take
the product of two numbers

00:01:23.860 --> 00:01:25.900
and you raise them
to some exponent,

00:01:25.900 --> 00:01:28.750
that's equivalent to
taking each of the numbers

00:01:28.750 --> 00:01:29.940
to that exponent.

00:01:29.940 --> 00:01:32.580
And then taking their product.

00:01:32.580 --> 00:01:34.530
And here I just used
the example with 4,

00:01:34.530 --> 00:01:41.020
but you could do this really
with any arbitrary-- actually

00:01:41.020 --> 00:01:43.460
any exponent.

00:01:43.460 --> 00:01:45.140
This property holds.

00:01:45.140 --> 00:01:47.850
And you could satisfy yourself
by trying different values,

00:01:47.850 --> 00:01:50.470
and using the same
logic right over here.

00:01:50.470 --> 00:01:52.680
But this is a general property.

00:01:52.680 --> 00:01:54.870
That-- let me
write it this way--

00:01:54.870 --> 00:02:02.125
that if I have a to
the b, to the c power,

00:02:02.125 --> 00:02:16.131
that this is going to be equal
to a to the c times b to the c

00:02:16.131 --> 00:02:16.630
power.

00:02:16.630 --> 00:02:18.440
And we'll use this to
throughout actually

00:02:18.440 --> 00:02:21.430
mathematics, when we try to
simplify things or rewrite

00:02:21.430 --> 00:02:23.700
an expression in
a different way.

00:02:23.700 --> 00:02:26.880
Now let me introduce you
to another core idea here.

00:02:26.880 --> 00:02:31.480
And this is the idea of raising
something to some power.

00:02:31.480 --> 00:02:33.630
And I'll just use example of 3.

00:02:33.630 --> 00:02:36.790
And then raising
that to some power.

00:02:36.790 --> 00:02:39.850
What could this
be simplified as?

00:02:39.850 --> 00:02:41.050
Well let's think about it.

00:02:41.050 --> 00:02:45.440
This is the same thing
as a to the third--

00:02:45.440 --> 00:02:49.300
let me copy and paste that--
as a to the third times

00:02:49.300 --> 00:02:50.010
a to the third.

00:02:54.330 --> 00:02:56.140
And what is a to the
third times-- So this

00:02:56.140 --> 00:02:58.390
is equal to a to the third
times a to the third.

00:02:58.390 --> 00:03:02.750
And that's going to be equal
to a to the 3 plus 3 power.

00:03:02.750 --> 00:03:05.760
We have the same
base, so we would add

00:03:05.760 --> 00:03:07.210
and they're being multiplied.

00:03:07.210 --> 00:03:09.690
They're being raised
to these two exponents.

00:03:09.690 --> 00:03:12.040
So it's going to be the
sum of the exponents, which

00:03:12.040 --> 00:03:16.700
of course is going to
be equal to a-- that's

00:03:16.700 --> 00:03:20.530
a different color a-- it's going
to be a to the sixth power.

00:03:23.980 --> 00:03:26.130
So what just happened over here?

00:03:26.130 --> 00:03:30.210
Well I took two a to the thirds.

00:03:30.210 --> 00:03:31.600
And I multiplied them together.

00:03:31.600 --> 00:03:36.360
So I took these two 3s
and added them together.

00:03:36.360 --> 00:03:39.200
So this essentially
right over here,

00:03:39.200 --> 00:03:41.935
you could view
this as 2 times 3.

00:03:47.230 --> 00:03:49.076
That's how we got the 6.

00:03:49.076 --> 00:03:50.700
When I raise something
to one exponent,

00:03:50.700 --> 00:03:53.110
and then raised it
to another, that's

00:03:53.110 --> 00:03:55.770
the equivalent to
raising the base

00:03:55.770 --> 00:03:58.700
to the product of
those two exponents.

00:03:58.700 --> 00:04:00.920
I just did it with this
example right over here.

00:04:00.920 --> 00:04:04.630
But I encourage you try other
numbers to see how this works.

00:04:04.630 --> 00:04:07.850
And I could to do
this in general.

00:04:07.850 --> 00:04:12.840
I could say a to the b power.

00:04:12.840 --> 00:04:17.220
And then-- let me
copy and paste that--

00:04:17.220 --> 00:04:21.506
and then I'm going to
raise that to the c power.

00:04:25.050 --> 00:04:26.564
Well what is that
going to give me?

00:04:26.564 --> 00:04:27.980
Well I'm essentially
going to have

00:04:27.980 --> 00:04:32.994
to take c of these,
so one, two, three.

00:04:32.994 --> 00:04:34.410
I don't know how
large of a number

00:04:34.410 --> 00:04:36.404
c is, so I'll just
do the dot, dot dot.

00:04:36.404 --> 00:04:38.780
So dot, dot, dot.

00:04:38.780 --> 00:04:41.835
I have c of these,
right over here.

00:04:47.810 --> 00:04:49.670
So what is that
going to be equal to?

00:04:49.670 --> 00:04:53.650
Well that is going to be
equal to a to the-- well

00:04:53.650 --> 00:04:55.575
for each of these
c, I'm going to have

00:04:55.575 --> 00:04:57.366
a b that I'm going
to add together.

00:04:57.366 --> 00:04:58.240
So let me write this.

00:04:58.240 --> 00:05:03.110
So I'm going to have a b plus
b plus b plus dot, dot, dot

00:05:03.110 --> 00:05:04.240
plus b.

00:05:04.240 --> 00:05:07.800
And now I have c
of these b's, so I

00:05:07.800 --> 00:05:10.300
have c b's right over here.

00:05:10.300 --> 00:05:13.530
Or you could view
this as a, this

00:05:13.530 --> 00:05:18.800
is equal to a to
the c times b power.

00:05:18.800 --> 00:05:23.380
c or a, you could do
a to the cb power.

00:05:23.380 --> 00:05:24.350
So very useful.

00:05:24.350 --> 00:05:32.740
So if someone were to say
what is 35 to the third power,

00:05:32.740 --> 00:05:37.760
and then that raised
to the seventh power?

00:05:37.760 --> 00:05:40.264
Well this is going to
obviously be a huge number.

00:05:40.264 --> 00:05:42.180
But we could at least
simplify the expression.

00:05:42.180 --> 00:05:46.840
This is going to be equal to
35 to the product of these two

00:05:46.840 --> 00:05:47.820
exponents.

00:05:47.820 --> 00:05:53.790
It's going to be 35 to the
3 times 7, or 35 to the 21,

00:05:53.790 --> 00:05:56.529
or to the 21st power.


Negative bases and exponents classification example

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.320
We're asked to
categorize the value

00:00:02.320 --> 00:00:04.220
of each exponential expression.

00:00:04.220 --> 00:00:05.960
And we categorize
each expression

00:00:05.960 --> 00:00:08.520
into either the bucket
less than negative 1,

00:00:08.520 --> 00:00:12.530
between negative 1 and
0, and between 0 and 1.

00:00:12.530 --> 00:00:14.350
And I encourage you
to pause this video

00:00:14.350 --> 00:00:17.440
and try this out on your own.

00:00:17.440 --> 00:00:19.630
So I assume you've
given a go at it.

00:00:19.630 --> 00:00:22.080
Now let's work
through it together.

00:00:22.080 --> 00:00:26.030
So I just copy and pasted
the exact same exercise right

00:00:26.030 --> 00:00:27.150
over here.

00:00:27.150 --> 00:00:29.220
And let's think about
each of these expressions.

00:00:29.220 --> 00:00:31.011
And actually, before
doing that, I actually

00:00:31.011 --> 00:00:33.820
want to draw a number line to
think about these categories.

00:00:33.820 --> 00:00:35.480
So it looks like
the important things

00:00:35.480 --> 00:00:37.880
to consider-- you have a 0.

00:00:37.880 --> 00:00:39.460
You have 1.

00:00:39.460 --> 00:00:41.660
And then you have negative 1.

00:00:41.660 --> 00:00:44.970
This first category right over
here, less than negative 1,

00:00:44.970 --> 00:00:48.560
that's all of these
values right over there.

00:00:48.560 --> 00:00:52.390
Then you have the category
between negative 1 and 0.

00:00:52.390 --> 00:00:56.250
That's these values
right over there.

00:00:56.250 --> 00:01:00.340
And then finally, you have
the category between 0 and 1.

00:01:00.340 --> 00:01:02.140
That's this category
right over here.

00:01:02.140 --> 00:01:04.709
So they didn't give us a
bucket for greater than 1,

00:01:04.709 --> 00:01:06.750
so I guess it's safe to
assume that none of these

00:01:06.750 --> 00:01:08.470
are going to be greater than 1.

00:01:08.470 --> 00:01:11.840
But let's go each expression
by each expression.

00:01:11.840 --> 00:01:17.560
So this first expression, 7
to the negative fourth power,

00:01:17.560 --> 00:01:19.550
we've already talked
about several times,

00:01:19.550 --> 00:01:22.220
that when you raise something
to a negative exponent,

00:01:22.220 --> 00:01:25.220
this isn't somehow negative 7
to the fourth power or anything

00:01:25.220 --> 00:01:25.840
like that.

00:01:27.864 --> 00:01:29.780
When you see a negative
number in an exponent,

00:01:29.780 --> 00:01:31.900
you should immediately
think reciprocal.

00:01:31.900 --> 00:01:35.130
So this is going to
be equal to 1 over 7

00:01:35.130 --> 00:01:37.374
to the positive fourth power.

00:01:37.374 --> 00:01:39.540
And you don't even have to
know what 7 to the fourth

00:01:39.540 --> 00:01:40.100
actually is.

00:01:40.100 --> 00:01:42.249
It's 2,401, if
you're interested.

00:01:42.249 --> 00:01:43.790
But you don't even
have to know that.

00:01:43.790 --> 00:01:45.415
You just know that
this right over here

00:01:45.415 --> 00:01:49.550
is going to be a large
positive number, so 1 over that

00:01:49.550 --> 00:01:51.390
is going to be a very
small positive number.

00:01:51.390 --> 00:01:54.220
It's going to be
between 0 and 1.

00:01:54.220 --> 00:01:58.150
This is actually one 1/2,401,
but you don't even have to know

00:01:58.150 --> 00:01:58.650
that.

00:01:58.650 --> 00:02:00.460
This is a large
number, and so you say,

00:02:00.460 --> 00:02:02.240
well, this is going
to get pretty close.

00:02:02.240 --> 00:02:04.550
It's positive, so it's
going to be above 0.

00:02:04.550 --> 00:02:06.310
But it's not going to
be too much above 0.

00:02:06.310 --> 00:02:07.820
It's going to sit on the
number line-- actually,

00:02:07.820 --> 00:02:09.710
I'm not even doing
justice right over there.

00:02:09.710 --> 00:02:12.110
The way I drew it, at
scale, it's not even

00:02:12.110 --> 00:02:15.780
a pixel to the right
of 0 right over there.

00:02:15.780 --> 00:02:20.790
So this is going to
be between 0 and 1.

00:02:20.790 --> 00:02:22.690
And actually, I'll
draw lines like this,

00:02:22.690 --> 00:02:26.170
to show roughly on our number
line, where it would sit.

00:02:26.170 --> 00:02:29.450
Now let's think about
negative 7 to the third power.

00:02:29.450 --> 00:02:34.400
So negative 7 to the
third power, well,

00:02:34.400 --> 00:02:37.470
this is the same thing
as negative 7 times

00:02:37.470 --> 00:02:40.730
negative 7 times negative 7.

00:02:40.730 --> 00:02:44.190
And you could also think
about this as negative 1 times

00:02:44.190 --> 00:02:49.300
7 times negative 1 times
7 times negative 1 times

00:02:49.300 --> 00:02:52.620
7, which is the
same thing as-- you

00:02:52.620 --> 00:02:55.050
have three negative 1's
being multiplied together--

00:02:55.050 --> 00:02:59.950
so negative 1 to the third power
times 7 to the third power.

00:02:59.950 --> 00:03:02.636
Well, what's negative
1 to the third power?

00:03:02.636 --> 00:03:04.260
Well, it's negative
1 times negative 1,

00:03:04.260 --> 00:03:06.801
which would be positive 1, but
then times another negative 1.

00:03:06.801 --> 00:03:08.630
So this is negative
1 right over here.

00:03:08.630 --> 00:03:11.850
So this is going to be equal
to-- let me write it this way.

00:03:11.850 --> 00:03:14.120
You could write it either
as-- I could write it

00:03:14.120 --> 00:03:17.620
like this-- negative
7 to the third power.

00:03:17.620 --> 00:03:19.032
So 7 to the third,
once again, we

00:03:19.032 --> 00:03:20.490
don't even have to
know what it is.

00:03:20.490 --> 00:03:23.426
This is going to be a
large positive value.

00:03:23.426 --> 00:03:24.550
But then we're negating it.

00:03:24.550 --> 00:03:27.370
If we had 7 to the third,
it would be greater than 1.

00:03:27.370 --> 00:03:29.700
But negative 7 to the
third is definitely

00:03:29.700 --> 00:03:31.670
less than negative 1.

00:03:31.670 --> 00:03:34.700
So this fits into that
bucket right over there.

00:03:34.700 --> 00:03:36.965
Now we have negative 7
to the negative fourth.

00:03:40.760 --> 00:03:43.210
So let me write it over
here-- so negative 7

00:03:43.210 --> 00:03:44.504
to the negative fourth.

00:03:44.504 --> 00:03:45.920
So this could be
a little daunting

00:03:45.920 --> 00:03:49.020
because you have a negative base
raised to a negative exponent.

00:03:49.020 --> 00:03:51.170
But whenever I see
a negative exponent,

00:03:51.170 --> 00:03:54.300
the first thing I worry about or
the first thing my brain wants

00:03:54.300 --> 00:03:56.480
to do is take a reciprocal.

00:03:56.480 --> 00:04:05.870
So this is going to be 1 over
negative 7 to the fourth power.

00:04:05.870 --> 00:04:08.670
And once again, the same
idea as we just saw over

00:04:08.670 --> 00:04:11.860
here, we could view this
as 1 over negative 1

00:04:11.860 --> 00:04:16.464
to the fourth power times 7
to the fourth power, which

00:04:16.464 --> 00:04:18.839
is going to be equal to--
negative 1 to the fourth power,

00:04:18.839 --> 00:04:20.450
this is just going
to be equal to 1.

00:04:20.450 --> 00:04:25.660
So this is going to be equal to
1 over 7 to the fourth power.

00:04:25.660 --> 00:04:27.730
Well, we've already
thought about what 1 over 7

00:04:27.730 --> 00:04:29.030
to the fourth power is.

00:04:29.030 --> 00:04:34.170
It is between 0 and 1.

00:04:34.170 --> 00:04:39.192
So this right over here
is between 0 and 1.

00:04:39.192 --> 00:04:40.650
Another way you
can think about it,

00:04:40.650 --> 00:04:43.250
this is going to be 1
over negative 7 times

00:04:43.250 --> 00:04:45.632
negative 7 times negative
7 times negative 7.

00:04:45.632 --> 00:04:48.090
You have a negative times a
negative times a negative times

00:04:48.090 --> 00:04:49.030
a negative.

00:04:49.030 --> 00:04:50.940
Well, that's going
to be a positive.

00:04:50.940 --> 00:04:54.150
So it's positive 7
to the fourth power.

00:04:54.150 --> 00:04:56.860
Now let's think about this one
right over here-- negative 7

00:04:56.860 --> 00:04:58.940
to the negative fifth power.

00:04:58.940 --> 00:05:01.790
So let me write this
out-- negative 7

00:05:01.790 --> 00:05:03.920
to the negative fifth power.

00:05:03.920 --> 00:05:06.530
Well once again, as soon as
you see that negative exponent,

00:05:06.530 --> 00:05:08.160
I want to just take
the reciprocal,

00:05:08.160 --> 00:05:09.993
so I don't have a
negative exponent anymore.

00:05:09.993 --> 00:05:14.800
So it's 1 over negative
7 to the fifth power.

00:05:14.800 --> 00:05:19.690
And we could rewrite
this as 1 over negative 1

00:05:19.690 --> 00:05:27.980
to the fifth times 7
to the fifth, which

00:05:27.980 --> 00:05:33.024
is equal to-- well, what's a
negative 1 to the fifth power?

00:05:33.024 --> 00:05:34.690
Well a negative 1
raised to an odd power

00:05:34.690 --> 00:05:36.250
is going to be a negative 1.

00:05:36.250 --> 00:05:41.710
So this is going to be negative
1 over 7 to the fifth power.

00:05:41.710 --> 00:05:44.507
So where is 1 over 7
to the fifth power?

00:05:44.507 --> 00:05:46.090
Well, that would be
really close to 0.

00:05:46.090 --> 00:05:47.190
This would be positive.

00:05:47.190 --> 00:05:50.470
1 over 7 to the fifth would be
a very small positive number.

00:05:50.470 --> 00:05:52.280
But now we're taking
the negative of that.

00:05:52.280 --> 00:05:55.410
So we're going to jump on
to the other side of 0.

00:05:55.410 --> 00:05:58.276
So this is going to be
between negative 1 and 0.

00:06:01.340 --> 00:06:05.040
And then finally, negative
7 to the fifth power,

00:06:05.040 --> 00:06:08.290
well that's just going
to be-- this, once again,

00:06:08.290 --> 00:06:10.720
you could view that
as negative 1 times

00:06:10.720 --> 00:06:15.650
7 to the fifth power, which is
the same thing as negative 1

00:06:15.650 --> 00:06:18.940
to the fifth times 7
to the fifth, which

00:06:18.940 --> 00:06:23.200
is the same thing as
negative 7 to the fifth.

00:06:23.200 --> 00:06:26.750
7 to the fifth is a very
large number, larger than 1.

00:06:26.750 --> 00:06:29.650
So the negative of that is going
to be a very negative number.

00:06:29.650 --> 00:06:31.580
It's going to be
less than negative 1.

00:06:31.580 --> 00:06:35.490
So this is going to be less
than negative 1 right over here.

00:06:35.490 --> 00:06:37.710
So now let's take our results.

00:06:37.710 --> 00:06:39.229
And since I have
a bad memory, I'm

00:06:39.229 --> 00:06:41.770
essentially probably going to
have to think through it again.

00:06:41.770 --> 00:06:43.561
And let's actually put
them in the buckets.

00:06:43.561 --> 00:06:45.250
7 to the negative
fourth power-- this

00:06:45.250 --> 00:06:46.750
is not going to be
a negative value.

00:06:46.750 --> 00:06:49.410
This is 1 over 7 to the fourth.

00:06:49.410 --> 00:06:52.360
This is going to
be between 0 and 1.

00:06:52.360 --> 00:06:56.910
Negative 7 to the third
power-- that's essentially,

00:06:56.910 --> 00:06:58.290
if you take 7 to
the third power,

00:06:58.290 --> 00:07:00.240
you get a very large
positive number.

00:07:00.240 --> 00:07:02.440
But then, since that's
going to be times negative 1

00:07:02.440 --> 00:07:03.940
to the third power,
it's going to be

00:07:03.940 --> 00:07:05.650
the negative version of that.

00:07:05.650 --> 00:07:08.020
So that's going to be
less than negative 1.

00:07:08.020 --> 00:07:10.520
Negative 7 to the
negative fourth-- well,

00:07:10.520 --> 00:07:13.900
we already determined,
especially because this ends up

00:07:13.900 --> 00:07:17.890
being 1 over negative
7 to the fourth power

00:07:17.890 --> 00:07:21.890
and that's an even exponent,
it ends up being the same as 7

00:07:21.890 --> 00:07:23.480
to the negative fourth.

00:07:23.480 --> 00:07:28.030
And then you had negative 7
to the negative fifth power.

00:07:28.030 --> 00:07:32.340
So once again, that was
going to be 1 over negative 7

00:07:32.340 --> 00:07:34.150
to the fifth power.

00:07:34.150 --> 00:07:38.130
And so we found that to be
between negative 1 and 0.

00:07:38.130 --> 00:07:41.427
And then finally,
this right over here,

00:07:41.427 --> 00:07:42.510
it's going to be negative.

00:07:42.510 --> 00:07:44.400
And it's going to be a
very negative number.

00:07:44.400 --> 00:07:45.970
It's less than negative 1.

00:07:45.970 --> 00:07:48.030
And so those are our choices.

00:07:48.030 --> 00:07:51.290
And we can check our
answer, and we got it right.


More negative exponent intuition

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:04.980
I don't think we can do enough
videos on why raising something

00:00:04.980 --> 00:00:09.050
to a negative exponent
is equivalent to 1

00:00:09.050 --> 00:00:17.390
over that base raised to
the positive exponent,

00:00:17.390 --> 00:00:18.310
I should say.

00:00:18.310 --> 00:00:22.200
And to get more intuition
about why this makes sense,

00:00:22.200 --> 00:00:24.940
I look at different
powers of 2, and then

00:00:24.940 --> 00:00:29.100
think about what makes sense
as we go to exponents below 0,

00:00:29.100 --> 00:00:32.150
integer exponents below 0.

00:00:32.150 --> 00:00:37.470
So let's start with
2 to the third power.

00:00:37.470 --> 00:00:39.120
Well, 2 to the third
power is 2 times

00:00:39.120 --> 00:00:42.250
2 times 2, which of
course is equal to 8.

00:00:42.250 --> 00:00:45.120
Now what about 2 to
the second power?

00:00:45.120 --> 00:00:47.100
Well that's going to
be 2 times 2, which

00:00:47.100 --> 00:00:49.240
is of course equal to 4.

00:00:49.240 --> 00:00:50.953
And to go from 2
to the third to 2

00:00:50.953 --> 00:00:52.850
to the second power,
what happened here?

00:00:52.850 --> 00:00:54.320
Well, we divided by 2.

00:00:56.860 --> 00:00:58.060
Now, let's keep going.

00:00:58.060 --> 00:01:00.570
What about 2 to the first power?

00:01:00.570 --> 00:01:02.570
Well, that's just
2, and once again,

00:01:02.570 --> 00:01:08.730
to go from 2 squared to 2 to the
first power, we divided by 2.

00:01:08.730 --> 00:01:10.985
Now things are going
to get interesting.

00:01:10.985 --> 00:01:13.470
2 to the 0-th power,
and actually this

00:01:13.470 --> 00:01:16.300
will help to build the
intuition of why it's something

00:01:16.300 --> 00:01:19.710
nonzero to the 0-th
power is defined to be 1.

00:01:19.710 --> 00:01:23.970
Well, so far, every time we
decremented the exponent by 1,

00:01:23.970 --> 00:01:28.630
we essentially divided by 2,
so we should divide by 2 again.

00:01:28.630 --> 00:01:31.559
So if we divide by
2 again, we get 1.

00:01:31.559 --> 00:01:32.975
And this is part
of the motivation

00:01:32.975 --> 00:01:36.730
of why 2 to the 0 power
should be equal to 1.

00:01:36.730 --> 00:01:38.740
But let's keep going.

00:01:38.740 --> 00:01:41.600
What should 2 to
the negative 1 power

00:01:41.600 --> 00:01:43.860
be, if we want to be
consistent about continuing

00:01:43.860 --> 00:01:45.530
to divide by 2?

00:01:45.530 --> 00:01:49.640
Well, we divide by
2 again, and so this

00:01:49.640 --> 00:01:52.320
is going to be equal to 1/2.

00:01:52.320 --> 00:01:54.690
Notice, 2 to the
negative 1 is 1/2.

00:01:54.690 --> 00:01:57.040
2 to the 1 is equal to 1.

00:01:57.040 --> 00:02:00.480
This is equal to the
reciprocal of this.

00:02:00.480 --> 00:02:01.320
Let's keep going.

00:02:01.320 --> 00:02:02.940
This is fun.

00:02:02.940 --> 00:02:06.290
So what should 2 to the
negative 2 power be?

00:02:06.290 --> 00:02:08.949
Well, we should
divide by 2 again.

00:02:08.949 --> 00:02:12.130
Divide by 2 again,
you get to 1/4.

00:02:12.130 --> 00:02:13.530
I think you see the pattern.

00:02:13.530 --> 00:02:18.080
2 to the negative 3 power, well
we should divide by 2 again.

00:02:18.080 --> 00:02:23.370
And we get to 1/8, which
is the reciprocal of 2

00:02:23.370 --> 00:02:25.110
to the third power.

00:02:25.110 --> 00:02:27.170
So once again,
another way to think

00:02:27.170 --> 00:02:33.010
about why negative exponents
are about taking reciprocals.

00:02:33.010 --> 00:02:36.730
Taking something to the negative
exponent is equivalent to 1

00:02:36.730 --> 00:02:41.063
over taking that same base
to the positive exponent.


Thinking more about negative exponents

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:02.620
In the last video,
we gave an argument

00:00:02.620 --> 00:00:06.140
for why 3 to the
negative 2 power

00:00:06.140 --> 00:00:11.310
should be equivalent to 1 over
3 to the positive 2 power.

00:00:11.310 --> 00:00:13.370
I now want to do a
very similar argument,

00:00:13.370 --> 00:00:17.210
but I want to do it for general
bases and general exponents.

00:00:17.210 --> 00:00:20.520
So in particular, I
want to think about what

00:00:20.520 --> 00:00:23.970
a to the b power,
or in particular,

00:00:23.970 --> 00:00:28.030
what a to the negative b
power should be equal to.

00:00:28.030 --> 00:00:31.600
So we know that we want
to uphold the property

00:00:31.600 --> 00:00:34.050
that, if we have the base
raised to an exponent

00:00:34.050 --> 00:00:37.370
and then that's being multiplied
by that same base raised

00:00:37.370 --> 00:00:39.230
to another exponent,
that we would just

00:00:39.230 --> 00:00:40.600
add the two exponents.

00:00:40.600 --> 00:00:43.760
So just to be clear, we want
to maintain this property,

00:00:43.760 --> 00:00:54.440
that a to the b power times
a to the negative b power

00:00:54.440 --> 00:01:01.630
should be equal to a to
the b plus negative b

00:01:01.630 --> 00:01:04.069
power, which of
course is going to be

00:01:04.069 --> 00:01:07.390
equal to a to the 0-th power.

00:01:07.390 --> 00:01:11.110
And as long as a
does not equal to 0,

00:01:11.110 --> 00:01:14.350
this should be equal to 1.

00:01:14.350 --> 00:01:18.730
Let me write that--
for a does not equal 0.

00:01:18.730 --> 00:01:24.280
So in general-- actually I'll
just copy and paste this-- a

00:01:24.280 --> 00:01:27.410
to the b times a to
the negative b-- copy

00:01:27.410 --> 00:01:31.065
and paste-- should
be equal to 1.

00:01:31.065 --> 00:01:35.111
So this right over here
needs to be equal to 1.

00:01:35.111 --> 00:01:37.610
And so if we want think about,
what does a to the negative b

00:01:37.610 --> 00:01:38.110
equal?

00:01:38.110 --> 00:01:39.850
We just divide both
sides by a to the b.

00:01:39.850 --> 00:01:43.075
And we would get a
to the negative b.

00:01:47.020 --> 00:01:49.515
I've kept the colors
consistent this long.

00:01:49.515 --> 00:01:50.920
Gotta keep going.

00:01:50.920 --> 00:01:55.280
a to the negative b is
equal to 1 over a to the b.

00:01:58.940 --> 00:02:01.510
And so I just want
to review this.

00:02:01.510 --> 00:02:03.350
And it's really
important to keep in mind

00:02:03.350 --> 00:02:05.800
because when you see
something as a base

00:02:05.800 --> 00:02:07.990
to a negative exponent,
there's a temptation

00:02:07.990 --> 00:02:09.940
to somehow introduce
a negative number.

00:02:09.940 --> 00:02:12.860
But a negative
exponent is really

00:02:12.860 --> 00:02:14.860
about taking the reciprocal.

00:02:14.860 --> 00:02:17.920
So you're taking the reciprocal
of this number raised

00:02:17.920 --> 00:02:19.885
to the positive base.

00:02:19.885 --> 00:02:21.260
And we've just
shown the argument

00:02:21.260 --> 00:02:22.940
that if we want this
property be true--

00:02:22.940 --> 00:02:25.030
and a lot of the
properties of exponents

00:02:25.030 --> 00:02:27.250
are all about just keeping
consistency, keeping

00:02:27.250 --> 00:02:31.180
the old properties continuing
to be true as we introduce

00:02:31.180 --> 00:02:33.500
new definitions and
new properties--

00:02:33.500 --> 00:02:35.820
and we see that if we
want this to be true,

00:02:35.820 --> 00:02:39.970
then a to the negative b
should be 1 over a to the b.

00:02:39.970 --> 00:02:42.180
And in other videos we'll
show other motivations

00:02:42.180 --> 00:02:47.120
for why a negative exponent is
essentially defined this way.

00:02:47.120 --> 00:02:49.070
But now that we've
seen this is true,

00:02:49.070 --> 00:02:53.540
for any non-zero
a and any b, let's

00:02:53.540 --> 00:02:54.870
actually do some examples.

00:02:54.870 --> 00:02:57.930
And actually, let's think about
the size of these numbers.

00:02:57.930 --> 00:03:03.810
So for example-- and let
me introduce new colors

00:03:03.810 --> 00:03:04.570
right over here.

00:03:08.010 --> 00:03:09.750
I said I would
introduce new colors,

00:03:09.750 --> 00:03:14.400
and I didn't-- if I were to
take 5 to the third power,

00:03:14.400 --> 00:03:16.850
we know that's 5
times 5 times 5.

00:03:16.850 --> 00:03:20.600
That is equal to 125.

00:03:20.600 --> 00:03:22.830
It's a reasonably large number.

00:03:22.830 --> 00:03:25.630
Now what should 5 to
the negative 3 power be?

00:03:25.630 --> 00:03:27.530
And I encourage you
to pause this video

00:03:27.530 --> 00:03:30.020
and think about this
before I tell you.

00:03:30.020 --> 00:03:31.370
Use what I just told you.

00:03:31.370 --> 00:03:34.590
What should 5 to the
negative 3 power be?

00:03:34.590 --> 00:03:37.160
Well, I assume you've
paused the video.

00:03:37.160 --> 00:03:44.270
So this is going to
be equal to 1 over 5

00:03:44.270 --> 00:03:53.570
to the positive 3 power,
which is equal to 1/125.

00:03:53.570 --> 00:03:55.290
So this is really interesting.

00:03:55.290 --> 00:03:59.530
5 to the third was a
reasonably large number.

00:03:59.530 --> 00:04:02.170
While 5 to the negative 3,
it wasn't a negative number.

00:04:02.170 --> 00:04:05.280
It's still a positive number,
but it's a very small number.

00:04:05.280 --> 00:04:08.270
It's 1/125.

00:04:08.270 --> 00:04:10.370
Let's do two more examples.

00:04:10.370 --> 00:04:21.160
Let's say I were to
take negative 1/2,

00:04:21.160 --> 00:04:24.540
and I were to raise
this to the third power.

00:04:24.540 --> 00:04:26.730
What is that going to be?

00:04:26.730 --> 00:04:31.790
Well, it's negative 1/2
times negative 1/2 times

00:04:31.790 --> 00:04:34.740
negative 1/2, which
is, of course,

00:04:34.740 --> 00:04:38.030
going to be equal
to negative 1/8.

00:04:41.250 --> 00:04:49.060
Now what should negative
1/2 to the negative 3 power

00:04:49.060 --> 00:04:51.330
be, based on everything
that we just talked about?

00:04:51.330 --> 00:04:53.740
And once again, I encourage
you to pause this video

00:04:53.740 --> 00:04:55.180
and think about it.

00:04:55.180 --> 00:05:02.230
Well it should be equal
to 1 over negative 1/2

00:05:02.230 --> 00:05:04.640
to the positive 3 power.

00:05:04.640 --> 00:05:06.360
Well, we just figured
out what that is.

00:05:06.360 --> 00:05:11.090
Negative 1/2 to the positive
3 power is negative 1/8.

00:05:11.090 --> 00:05:13.170
This thing right over
here should be equal

00:05:13.170 --> 00:05:14.770
to negative 1/8.

00:05:14.770 --> 00:05:20.880
So it's going to be equal
to 1 over negative 1/8.

00:05:20.880 --> 00:05:22.490
And what's that equal to?

00:05:22.490 --> 00:05:26.750
Well that's equal to
1 times the reciprocal

00:05:26.750 --> 00:05:28.130
of this denominator over here.

00:05:28.130 --> 00:05:33.290
So it's 1 times
negative 8/1, which

00:05:33.290 --> 00:05:36.690
is going to be
equal to negative 8.

00:05:36.690 --> 00:05:38.120
So this is interesting.

00:05:38.120 --> 00:05:43.920
Normally when you take a
number whose absolute value is

00:05:43.920 --> 00:05:46.634
less than 1 and you take it to
larger and larger exponents,

00:05:46.634 --> 00:05:48.300
it gets smaller and
smaller and smaller.

00:05:48.300 --> 00:05:51.930
When you multiply either a
positive or negative 1/2 times

00:05:51.930 --> 00:05:55.640
1/2 times 1/2, its magnitude,
its absolute value,

00:05:55.640 --> 00:05:58.170
is normally going to get
smaller and smaller and smaller.

00:05:58.170 --> 00:06:00.490
But now, when you raise
it to a negative exponent,

00:06:00.490 --> 00:06:04.780
it's absolute value is getting
larger and larger and larger.

00:06:04.780 --> 00:06:08.840
And that's because, once again,
you are taking the reciprocal.

00:06:08.840 --> 00:06:12.324
The important thing I
want to emphasize here

00:06:12.324 --> 00:06:13.990
is that you shouldn't
view this negative

00:06:13.990 --> 00:06:16.010
as when you think about
negative multiplication.

00:06:16.010 --> 00:06:18.460
Notice, this negative
did not change

00:06:18.460 --> 00:06:22.200
the sign of what the
eventual value is.

00:06:22.200 --> 00:06:23.380
You've got negative 1/8.

00:06:23.380 --> 00:06:24.630
You got negative 8.

00:06:24.630 --> 00:06:29.130
What it did is, it
changed the magnitude.

00:06:29.130 --> 00:06:30.980
So this is negative 1/8.

00:06:30.980 --> 00:06:33.000
When you took to the
negative third power,

00:06:33.000 --> 00:06:37.430
you took the reciprocal of
that, which is negative 8.


Introduction to negative exponents

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.100
Let's review a little bit of
what we know about exponents.

00:00:04.100 --> 00:00:10.660
So if I were to have 3 to
the second power times 3

00:00:10.660 --> 00:00:17.130
to the third power,
what does that equal?

00:00:17.130 --> 00:00:19.390
Well, we could expand
it out just as a review.

00:00:19.390 --> 00:00:22.705
3 to the second power-- well,
that's going to be 3 times 3.

00:00:25.310 --> 00:00:27.040
And then 3 to the
third power-- we're

00:00:27.040 --> 00:00:29.870
going to multiply it by
3 to the third power--

00:00:29.870 --> 00:00:34.910
is going to be 3
times 3 times 3.

00:00:34.910 --> 00:00:36.720
So what is this
going to be equal to?

00:00:36.720 --> 00:00:41.520
Well, this is going to be equal
to 3 to the 2 plus 3 power.

00:00:48.920 --> 00:00:51.600
And this comes straight
out of what an exponent is.

00:00:51.600 --> 00:00:54.130
We're taking two 3's and
multiplying it together.

00:00:54.130 --> 00:00:56.510
And then we're going to
multiply that with three 3's

00:00:56.510 --> 00:00:57.890
being multiplied together.

00:00:57.890 --> 00:01:01.150
So you're going to end up
with 2 plus 3, or five,

00:01:01.150 --> 00:01:02.890
3's being multiplied together.

00:01:02.890 --> 00:01:07.490
So this, of course, is
going to be equal to 3

00:01:07.490 --> 00:01:08.285
to the fifth power.

00:01:12.474 --> 00:01:13.890
And that's a little
bit of review.

00:01:13.890 --> 00:01:17.290
And this is one of the
core exponent properties.

00:01:17.290 --> 00:01:21.750
If you have the same base
multiplied to exponents,

00:01:21.750 --> 00:01:24.630
and you're taking the product
of those two expressions,

00:01:24.630 --> 00:01:27.610
you can essentially, or you
do, just add the exponents.

00:01:27.610 --> 00:01:30.780
3 to some exponent times
3 to some other exponent

00:01:30.780 --> 00:01:35.047
is going to be 3 to the
sum of those two exponents.

00:01:35.047 --> 00:01:36.630
So that's kind of
interesting, but now

00:01:36.630 --> 00:01:39.630
let's use this property
right over here

00:01:39.630 --> 00:01:43.430
to try to think about what a
negative exponent should be.

00:01:43.430 --> 00:01:50.480
So we're going to try to think
about what should-- let's just

00:01:50.480 --> 00:01:56.122
say 3 to the negative 2 power--
what should this be equal to?

00:01:56.122 --> 00:01:58.580
And I encourage you to pause
this video and think about it.

00:01:58.580 --> 00:02:01.030
Think about, what should
this be equal to if we

00:02:01.030 --> 00:02:03.140
want this property
to hold, that if you

00:02:03.140 --> 00:02:06.759
have the same base raised
to different exponents,

00:02:06.759 --> 00:02:09.050
and you're taking the product
of those two expressions,

00:02:09.050 --> 00:02:13.820
that you can take that base to
the sum of those two exponents.

00:02:13.820 --> 00:02:16.840
Well, let's try a
little experiment here.

00:02:16.840 --> 00:02:28.220
What should 3 squared times
3 to the negative 2 power be?

00:02:28.220 --> 00:02:30.590
What should this be equal to?

00:02:30.590 --> 00:02:33.280
Well, if we want to keep
this property going,

00:02:33.280 --> 00:02:40.280
this should be equal
to 3 to the 2 plus

00:02:40.280 --> 00:02:48.220
negative 2 power, which, of
course, would be equal to 3

00:02:48.220 --> 00:02:49.725
to the 0-th power.

00:02:52.720 --> 00:02:55.180
And any non-zero number
to the 0-th power

00:02:55.180 --> 00:02:57.910
is going to be equal to 1.

00:02:57.910 --> 00:03:00.660
So this should be equal to 1.

00:03:00.660 --> 00:03:03.540
So we should define 3 to
the negative 2 in some way

00:03:03.540 --> 00:03:06.080
so that 3 squared times
3 to the negative 2

00:03:06.080 --> 00:03:07.890
should be equal to 1.

00:03:07.890 --> 00:03:12.740
We need to do that if we want
this property to hold right

00:03:12.740 --> 00:03:13.670
over there.

00:03:13.670 --> 00:03:15.840
So let's just think about
what 3 to the negative 2

00:03:15.840 --> 00:03:16.910
should actually be.

00:03:16.910 --> 00:03:19.530
So we want this to equal 1.

00:03:19.530 --> 00:03:23.745
Let me copy and paste that
so that we can manipulate it

00:03:23.745 --> 00:03:24.870
a little bit algebraically.

00:03:29.651 --> 00:03:30.150
All right.

00:03:30.150 --> 00:03:31.320
So we have that on one side.

00:03:31.320 --> 00:03:36.080
We want that to be equal to 1.

00:03:36.080 --> 00:03:38.250
And let me clean
this up a little bit.

00:03:38.250 --> 00:03:40.480
We want that to be equal to 1.

00:03:40.480 --> 00:03:42.260
So to define 3 to
the negative 2,

00:03:42.260 --> 00:03:44.640
or to think about what 3 to
the negative 2 should be,

00:03:44.640 --> 00:03:49.300
let's divide both
sides by 3 squared.

00:03:49.300 --> 00:03:51.675
Let's divide both sides by that.

00:03:51.675 --> 00:03:54.160
So let me copy that.

00:03:54.160 --> 00:03:56.460
So we're going to divide
this side by that,

00:03:56.460 --> 00:04:00.260
and we're going to
divide this side by that.

00:04:00.260 --> 00:04:03.617
And let me draw a line so that
you know that we're dividing.

00:04:03.617 --> 00:04:05.200
So we're going to
divide the left hand

00:04:05.200 --> 00:04:08.130
by that and the
right hand by that.

00:04:08.130 --> 00:04:11.190
Well, 3 squared over 3
squared-- those cancel out.

00:04:11.190 --> 00:04:13.540
And now we get a really
interesting result.

00:04:13.540 --> 00:04:20.060
We get that 3 to
the negative 2 power

00:04:20.060 --> 00:04:27.540
should be equal to 1 over
3 to the second power.

00:04:32.750 --> 00:04:34.960
And I really want you to
internalize this right over

00:04:34.960 --> 00:04:35.110
here.

00:04:35.110 --> 00:04:37.040
You have something to
the negative power,

00:04:37.040 --> 00:04:39.230
to the negative 2
power, in this case.

00:04:39.230 --> 00:04:44.110
That's equal to 1 over 3
to the positive version

00:04:44.110 --> 00:04:45.930
of that exponent.

00:04:45.930 --> 00:04:47.630
And this is true in general.

00:04:47.630 --> 00:04:50.650
In the next video, we'll
use a very similar argument

00:04:50.650 --> 00:04:55.650
to prove this for any non-zero
base and any exponent,

00:04:55.650 --> 00:04:58.060
that raising it to
the negative should

00:04:58.060 --> 00:05:00.209
be-- if you want this
exponent property to hold,

00:05:00.209 --> 00:05:02.000
that you would add the
exponents when we're

00:05:02.000 --> 00:05:03.750
taking the product of
the same base raised

00:05:03.750 --> 00:05:05.040
to different exponents.

00:05:05.040 --> 00:05:06.680
If you want that
property to hold,

00:05:06.680 --> 00:05:09.000
when you raise something
to the negative exponent,

00:05:09.000 --> 00:05:11.460
it should be 1 over
that base raised

00:05:11.460 --> 00:05:14.570
to the positive version
of that exponent.

00:05:14.570 --> 00:05:17.250
So 3 to the negative
2 power is 1 over 3

00:05:17.250 --> 00:05:20.700
squared, which is equal to 1/9.


How far can you see from a plane window

WEBVTT
Kind: captions
Language: en

00:00:00.344 --> 00:00:02.260
What I want to do in
this video is think about

00:00:02.260 --> 00:00:06.710
how far can I see from
the window of a plane

00:00:06.710 --> 00:00:09.470
when a plane is at
its cruising altitude.

00:00:09.470 --> 00:00:11.810
And just to give us some
numbers to work with,

00:00:11.810 --> 00:00:14.550
let's just assume that the
cruising altitude that we

00:00:14.550 --> 00:00:20.280
are working with, the altitude
of the plane, is 10 kilometers.

00:00:20.280 --> 00:00:23.630
And the number for the radius
of the earth-- and so this

00:00:23.630 --> 00:00:25.050
will be important
for figuring out

00:00:25.050 --> 00:00:29.410
how far I can see
from the plane.

00:00:29.410 --> 00:00:31.440
Let me do this is
a different color.

00:00:31.440 --> 00:00:39.490
The earth's radius-- I'll do
it in green-- in kilometers

00:00:39.490 --> 00:00:43.140
is 6,371 kilometers.

00:00:43.140 --> 00:00:46.472
And it's actually not
constant throughout the earth,

00:00:46.472 --> 00:00:48.930
but we're going to assume that
for the sake of this problem

00:00:48.930 --> 00:00:50.320
that Earth is a perfect sphere.

00:00:50.320 --> 00:00:51.820
It actually is not
one, but we will

00:00:51.820 --> 00:00:53.790
assume that it's
a perfect sphere.

00:00:53.790 --> 00:00:59.380
And its radius is
roughly 6,371 kilometers.

00:00:59.380 --> 00:01:01.670
So given this
information, can we

00:01:01.670 --> 00:01:05.200
figure out how far can we
see on a perfectly clear day

00:01:05.200 --> 00:01:07.900
from the window of
a plane cruising

00:01:07.900 --> 00:01:09.636
at an altitude of 10 kilometers?

00:01:09.636 --> 00:01:11.260
And I encourage you
to pause this video

00:01:11.260 --> 00:01:14.050
and try to think
about it on your own.

00:01:14.050 --> 00:01:15.680
So I assume you've
given a go at it.

00:01:15.680 --> 00:01:18.350
So let's draw Earth.

00:01:18.350 --> 00:01:20.220
And this by no stretch
of the imaginations

00:01:20.220 --> 00:01:24.370
will be a scale drawing either
of Earth or of the plane

00:01:24.370 --> 00:01:26.710
flying above the
surface of Earth.

00:01:26.710 --> 00:01:30.110
So let's say that this
right over here is Earth.

00:01:30.110 --> 00:01:33.780
I would want to draw it as a
perfect sphere, but I have not.

00:01:33.780 --> 00:01:36.860
But you understand
my limitations.

00:01:36.860 --> 00:01:38.230
Now let's see what we could do.

00:01:38.230 --> 00:01:43.080
So the plane is traveling at
an altitude of 10 kilometers.

00:01:43.080 --> 00:01:45.320
So let's say that this is
the plane right over here.

00:01:45.320 --> 00:01:47.460
And this distance, once
again, by no stretch

00:01:47.460 --> 00:01:51.785
of the imagination drawn
to scale, is 10 kilometers.

00:01:55.010 --> 00:01:57.100
And then if we were
to keep going down

00:01:57.100 --> 00:02:03.010
to the center of the
earth, that would

00:02:03.010 --> 00:02:09.180
be another 6,371 kilometers.

00:02:09.180 --> 00:02:12.920
And when you think about
how far you could see,

00:02:12.920 --> 00:02:15.660
your line of sight is
really forming a tangent

00:02:15.660 --> 00:02:18.740
with some point on the
surface of the earth.

00:02:18.740 --> 00:02:19.840
So this is me in my plane.

00:02:19.840 --> 00:02:21.673
And let's say I'm looking
at that direction.

00:02:21.673 --> 00:02:23.240
How far could I see?

00:02:23.240 --> 00:02:26.410
Well, I'm going to be
hitting the surface

00:02:26.410 --> 00:02:28.210
of the earth at some point.

00:02:28.210 --> 00:02:31.140
And that line of
sight is essentially

00:02:31.140 --> 00:02:33.620
going to be tangent
to the surface.

00:02:33.620 --> 00:02:36.951
So let me see if I can draw
it a little bit cleaner.

00:02:36.951 --> 00:02:39.200
So let's see, it's going to
be tangent to the surface.

00:02:39.200 --> 00:02:43.840
So it's going to look
something like that.

00:02:43.840 --> 00:02:45.960
And so this is the
furthest point I could see,

00:02:45.960 --> 00:02:47.500
the point at which
my line of sight

00:02:47.500 --> 00:02:50.620
is just tangent to the
surface of the earth.

00:02:50.620 --> 00:02:52.371
Well, how far is that
point of the surface

00:02:52.371 --> 00:02:54.078
going to be from the
center of the earth?

00:02:54.078 --> 00:02:56.540
Well, we're assuming that
Earth is a perfect sphere.

00:02:56.540 --> 00:03:07.410
So that's also going
to be 6,371 kilometers.

00:03:07.410 --> 00:03:11.510
And the other thing we
know is that a tangent line

00:03:11.510 --> 00:03:14.340
is going to be
perpendicular to a radius.

00:03:14.340 --> 00:03:17.450
So this right over here is
going to be a right angle.

00:03:17.450 --> 00:03:19.820
And all sorts of bells should
be ringing in your brain

00:03:19.820 --> 00:03:21.570
at this point
because we have just

00:03:21.570 --> 00:03:24.000
constructed a right triangle.

00:03:24.000 --> 00:03:27.240
And what we want to figure
out, how far we can see-- well,

00:03:27.240 --> 00:03:29.970
that's this unknown side
of this right triangle,

00:03:29.970 --> 00:03:32.490
the side that I
have marked in x.

00:03:32.490 --> 00:03:35.710
So we can just use the
Pythagorean theorem.

00:03:35.710 --> 00:03:39.570
We know that the square of
this side, so x squared,

00:03:39.570 --> 00:03:43.250
plus the square of this
other non-hypotenuse side,

00:03:43.250 --> 00:03:48.390
so 6,371 squared, is going
to be equal to the square

00:03:48.390 --> 00:03:50.074
of the hypotenuse.

00:03:50.074 --> 00:03:51.740
So what is the length
of the hypotenuse?

00:03:51.740 --> 00:03:56.620
Well, the hypotenuse is 6,371
kilometers plus this extra 10

00:03:56.620 --> 00:04:04.020
kilometers of altitude squared.

00:04:04.020 --> 00:04:14.490
And so we get x squared
plus 6,371 squared

00:04:14.490 --> 00:04:16.790
is equal to-- what
is this going to be?

00:04:16.790 --> 00:04:20.190
This is going to be this whole
distance right over here.

00:04:20.190 --> 00:04:26.720
The whole distance is just
going to be 6,381 kilometers,

00:04:26.720 --> 00:04:28.590
and we're going to square that.

00:04:28.590 --> 00:04:33.760
So x squared-- we just subtract
6,371 squared from both sides.

00:04:33.760 --> 00:04:39.310
We get x squared
is equal to 6,381

00:04:39.310 --> 00:04:45.190
squared minus 6,371 squared.

00:04:45.190 --> 00:04:50.715
Or we could say that x is
equal to the square root of all

00:04:50.715 --> 00:04:51.530
of this business.

00:04:51.530 --> 00:04:53.420
And I could just
copy and paste that.

00:04:53.420 --> 00:04:57.700
So copy, and paste.

00:04:57.700 --> 00:04:59.820
Let's throw that right in there.

00:04:59.820 --> 00:05:02.080
Actually, my radical
is not big enough.

00:05:02.080 --> 00:05:06.520
So let me make my radical
a little bit bigger.

00:05:06.520 --> 00:05:10.740
And then we can use a calculator
to actually compute this.

00:05:10.740 --> 00:05:13.142
So let me get my calculator out.

00:05:13.142 --> 00:05:15.100
And this is exciting
because now we can finally

00:05:15.100 --> 00:05:18.430
know on a perfectly clear
day how far we can see out

00:05:18.430 --> 00:05:19.690
of a plane.

00:05:19.690 --> 00:05:26.440
So it's going to be the
square root of 6,381 squared

00:05:26.440 --> 00:05:32.790
minus 6,371 squared, which
is going to be-- we deserve

00:05:32.790 --> 00:05:41.130
a drum roll-- roughly
357 kilometers,

00:05:41.130 --> 00:05:43.160
if we round to the
nearest kilometer.

00:05:43.160 --> 00:05:48.600
So x is approximately
equal to 357 kilometers

00:05:48.600 --> 00:05:49.570
in any direction.

00:05:49.570 --> 00:05:53.270
So you could see 357
kilometers in that direction.

00:05:53.270 --> 00:05:57.980
You could see 357 kilometers
in that direction.

00:05:57.980 --> 00:06:02.977
So let's say you're
looking at a map of the US,

00:06:02.977 --> 00:06:04.435
and your plane is
flying across it.

00:06:04.435 --> 00:06:05.476
And this is kind of neat.

00:06:08.050 --> 00:06:11.080
And once again, this is by
no means drawn to scale,

00:06:11.080 --> 00:06:14.515
but let's say that this right
over here is North America.

00:06:14.515 --> 00:06:16.255
That's the Yucatan Peninsula.

00:06:16.255 --> 00:06:17.340
That's Louisiana.

00:06:17.340 --> 00:06:19.256
I'll draw Lake Pontchartrain
right over there.

00:06:19.256 --> 00:06:20.750
And then see, that's Florida.

00:06:20.750 --> 00:06:22.780
And then that's the
rest of North America.

00:06:22.780 --> 00:06:24.360
That's California,
just like that.

00:06:24.360 --> 00:06:25.570
That's Vancouver.

00:06:25.570 --> 00:06:28.074
I could keep going--
Alaska, so on and so forth.

00:06:28.074 --> 00:06:29.490
So if you're flying
in your plane,

00:06:29.490 --> 00:06:32.740
and your plane is right here,
if we assume the numbers we just

00:06:32.740 --> 00:06:36.480
assumed, in any direction,
you can see 357 kilometers.

00:06:36.480 --> 00:06:41.290
So 357 kilometers would
be about that far.

00:06:41.290 --> 00:06:44.910
So you could see that
far in every direction.

00:06:44.910 --> 00:06:47.512
So it's kind of like seeing
an entire state-- or actually

00:06:47.512 --> 00:06:49.470
multiple states, depending
on what state you're

00:06:49.470 --> 00:06:52.039
thinking about-- of the
US from your window,

00:06:52.039 --> 00:06:53.580
if you could look
in every direction.

00:06:53.580 --> 00:06:54.390
Obviously, if
you're only sitting

00:06:54.390 --> 00:06:55.510
on one side of the
plane, you could only

00:06:55.510 --> 00:06:56.110
look out of one window.

00:06:56.110 --> 00:06:57.930
But maybe you could go switch
to the other side of the plane

00:06:57.930 --> 00:06:59.263
and see the other side, as well.

00:06:59.263 --> 00:07:02.710
But you're actually
seeing a substantive area

00:07:02.710 --> 00:07:05.430
when you look out of the window
from 10 kilometers, which

00:07:05.430 --> 00:07:10.750
is roughly about
32,000 feet in the air.


Constructing a geometric series for new users

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.700
A new social media site
boasts that its user base

00:00:03.700 --> 00:00:08.080
has increased 47% each
month for the past year.

00:00:08.080 --> 00:00:13.926
The number of users on January
1st of last year was 50,000.

00:00:13.926 --> 00:00:15.800
Which expression below
gives the total number

00:00:15.800 --> 00:00:18.430
of new users in thousands
that were added through month

00:00:18.430 --> 00:00:21.124
n of the past year, where 1 is
less than or equal to n, which

00:00:21.124 --> 00:00:22.290
is less than or equal to 12.

00:00:22.290 --> 00:00:24.570
And they give us some
choices of expressions

00:00:24.570 --> 00:00:26.900
for the total
number of new users

00:00:26.900 --> 00:00:29.960
that were added through month n.

00:00:29.960 --> 00:00:32.090
And I encourage you to
now pause this video

00:00:32.090 --> 00:00:34.250
and try to think about
which of these expressions

00:00:34.250 --> 00:00:38.121
actually show that,
that gives that value.

00:00:38.121 --> 00:00:40.620
Well to tackle it, I'm going
to make a little bit of a table

00:00:40.620 --> 00:00:41.510
here.

00:00:41.510 --> 00:00:50.060
So let's say we have month, and
then we have starting users--

00:00:50.060 --> 00:00:53.110
so users at the
start-- and then let's

00:00:53.110 --> 00:00:55.150
think about the users added.

00:00:55.150 --> 00:00:57.810
I want to give myself
some space to work with.

00:00:57.810 --> 00:00:59.960
And then users at
the end of the month.

00:01:02.720 --> 00:01:08.740
So in month 1, which is
January we can assume,

00:01:08.740 --> 00:01:10.464
we started with 50,000 users.

00:01:10.464 --> 00:01:12.380
They want us to write
the expression in 1,000.

00:01:12.380 --> 00:01:13.940
So we started with 50,000 users.

00:01:13.940 --> 00:01:15.230
And how many did we add?

00:01:15.230 --> 00:01:18.530
Well we added 47% of 50,000.

00:01:18.530 --> 00:01:22.160
So 50 times 47%.

00:01:22.160 --> 00:01:24.700
So times 0.47.

00:01:24.700 --> 00:01:26.270
So how many do we end with?

00:01:26.270 --> 00:01:29.980
Well 50 plus 50
times 0.47, that's

00:01:29.980 --> 00:01:37.120
going to be 50 times-- I'm gonna
do the 50 in green-- that's

00:01:37.120 --> 00:01:43.070
going to be 50 times 1.47.

00:01:43.070 --> 00:01:45.760
If this isn't clear,
just think about this.

00:01:45.760 --> 00:01:47.550
This you could
rewrite as 50 times 1.

00:01:47.550 --> 00:01:51.160
So 50 times 1 plus
50 times 0.47,

00:01:51.160 --> 00:01:57.680
that's going to be
50 times 1 plus 0.47.

00:01:57.680 --> 00:01:59.070
Or 1.47.

00:01:59.070 --> 00:02:01.640
So it's going to be this
thing right over here.

00:02:01.640 --> 00:02:04.970
So now let's go to month 2.

00:02:04.970 --> 00:02:08.080
We start with what we
ended the last month.

00:02:08.080 --> 00:02:10.419
So I could just copy
and paste this actually.

00:02:10.419 --> 00:02:11.880
Let me just do that.

00:02:11.880 --> 00:02:15.770
So copy and paste.

00:02:15.770 --> 00:02:17.410
So that's what we start with.

00:02:17.410 --> 00:02:19.540
Now what are we going to add?

00:02:19.540 --> 00:02:23.040
Well we're going to add
this, what we started with,

00:02:23.040 --> 00:02:30.789
times 0.47.

00:02:30.789 --> 00:02:32.330
And so what are we
going to end with?

00:02:32.330 --> 00:02:36.520
Well if you sum these two-- and
you could write it this way--

00:02:36.520 --> 00:02:48.680
this is going to be
this thing times 1.47.

00:02:48.680 --> 00:02:55.280
Or we could just write this
is 50 times 1.47 squared.

00:02:55.280 --> 00:02:57.390
And you might start
seeing a pattern now.

00:02:57.390 --> 00:02:59.310
Let's go to month 3.

00:02:59.310 --> 00:03:02.130
So month 3: what do we
start the month with?

00:03:02.130 --> 00:03:05.370
We start the month
with this thing.

00:03:05.370 --> 00:03:07.640
Let me copy and paste this.

00:03:07.640 --> 00:03:10.410
So, copy and paste.

00:03:10.410 --> 00:03:12.570
We start with that.

00:03:12.570 --> 00:03:13.425
What do we add?

00:03:13.425 --> 00:03:15.660
Well we're going to
take that, and we're

00:03:15.660 --> 00:03:19.340
going to multiply it times 47%.

00:03:19.340 --> 00:03:23.674
We're going to
multiply it times 0.47.

00:03:23.674 --> 00:03:25.340
And so what are we
going to end up with?

00:03:25.340 --> 00:03:31.180
We're gonna have this times
1 plus this times 0.47.

00:03:31.180 --> 00:03:41.800
That is going to be equal
to that times 1.471.

00:03:41.800 --> 00:03:47.430
Or we could just write this as
50 times 1.47 to the 3rd power.

00:03:47.430 --> 00:03:49.010
So what's the pattern here?

00:03:49.010 --> 00:03:51.620
Well in each month, we're
going to be starting

00:03:51.620 --> 00:03:56.710
with 50 times 1.47 to a
power 1 less than the month.

00:03:56.710 --> 00:03:58.410
In the third month,
the power here is 2.

00:03:58.410 --> 00:04:00.400
In the second month,
the power here is 1.

00:04:00.400 --> 00:04:03.670
In the first month, the
power here-- you don't see it

00:04:03.670 --> 00:04:09.350
but you can view this as
times 1.47 to the 0 power.

00:04:09.350 --> 00:04:10.770
So 1st month, 0 power.

00:04:10.770 --> 00:04:14.900
Third month, you
have the 1st power.

00:04:14.900 --> 00:04:18.494
Third month, you have the
second power over here.

00:04:18.494 --> 00:04:19.910
So if we're thinking
about the nth

00:04:19.910 --> 00:04:31.620
month this is going to
be 50 times 1.47 to the n

00:04:31.620 --> 00:04:37.384
minus 1th power, is what we're
going to start the month with.

00:04:37.384 --> 00:04:39.300
Now what are we going
to add in the nth month?

00:04:39.300 --> 00:04:42.050
Well it's going to
be that times 47%.

00:04:42.050 --> 00:04:44.660
So it's going to be--
we'll just copy and paste

00:04:44.660 --> 00:04:50.360
that-- so it's going
to be that times 47%.

00:04:54.460 --> 00:04:57.494
Times 0.47.

00:04:57.494 --> 00:04:59.160
And then, what are
we going to end with?

00:04:59.160 --> 00:05:01.080
Well when you add
these two things,

00:05:01.080 --> 00:05:04.949
you are going to get
50-- I'll just do it

00:05:04.949 --> 00:05:07.240
in the right colors instead
of copying and pasting it--

00:05:07.240 --> 00:05:18.035
you're going to get 50
times 1.47 to the nth power.

00:05:18.035 --> 00:05:19.660
So let's think about
how we can come up

00:05:19.660 --> 00:05:22.590
with the expression for the
total number of new users

00:05:22.590 --> 00:05:27.340
in thousands that were
added through month n.

00:05:27.340 --> 00:05:29.430
So there's a couple of
ways to think about it.

00:05:29.430 --> 00:05:32.480
You could say, well how
many total new users

00:05:32.480 --> 00:05:34.870
did we have at the
end of month n?

00:05:34.870 --> 00:05:37.620
Well at the end of month
n, we had that many.

00:05:37.620 --> 00:05:40.130
And then how many did we have
at the beginning of the year?

00:05:40.130 --> 00:05:41.630
Well we have 50,000.

00:05:41.630 --> 00:05:45.400
So how many total new users
did we add through month n.

00:05:45.400 --> 00:05:49.190
So we finished with this much.

00:05:49.190 --> 00:05:51.260
Let me just write it down.

00:05:51.260 --> 00:05:56.200
So we just finished
with that much.

00:05:58.720 --> 00:06:00.020
And let me paste that.

00:06:00.020 --> 00:06:01.740
So that's what we finished with.

00:06:01.740 --> 00:06:07.050
And we started
with 50,000 users.

00:06:07.050 --> 00:06:09.750
So this is essentially how
many we added through month n.

00:06:09.750 --> 00:06:12.360
Now do any of our
expressions look like that?

00:06:12.360 --> 00:06:13.950
Well, no not quite.

00:06:13.950 --> 00:06:16.220
If this one had a minus
50 right over here,

00:06:16.220 --> 00:06:18.790
if that said minus 50, then
that would've done the trick,

00:06:18.790 --> 00:06:20.570
but this doesn't do it.

00:06:20.570 --> 00:06:23.484
And none of the others really
seem to either have this form

00:06:23.484 --> 00:06:24.900
or seem to be
something that would

00:06:24.900 --> 00:06:27.600
be very easy to
manipulate in this form.

00:06:27.600 --> 00:06:30.240
So that's one way to do it, but
that's not one of our choices.

00:06:30.240 --> 00:06:32.120
So what's another way
of thinking about it.

00:06:32.120 --> 00:06:34.940
Well, we could literally
just add how many new users

00:06:34.940 --> 00:06:36.460
we added month by month.

00:06:36.460 --> 00:06:39.700
So we literally could just
add all of these things

00:06:39.700 --> 00:06:40.560
right over here.

00:06:45.850 --> 00:06:51.480
So let's see, we could literally
just add all of these terms.

00:06:51.480 --> 00:06:53.150
But let me simplify
it a little bit.

00:06:53.150 --> 00:06:55.790
So what are some common factors
that we see in all of these?

00:06:55.790 --> 00:06:59.410
Well we see they all have a 50.

00:06:59.410 --> 00:07:02.480
And they all have something
being multiplied by 0.47.

00:07:02.480 --> 00:07:07.520
So let's factor out
a 0.47 and a 50.

00:07:07.520 --> 00:07:08.665
So let's factor that out.

00:07:08.665 --> 00:07:10.290
So this is going to
be equal to-- if we

00:07:10.290 --> 00:07:19.000
were to sum all this up-- it's
going to be 0.47 times 50.

00:07:19.000 --> 00:07:20.550
And then what's left over?

00:07:20.550 --> 00:07:23.420
Times-- so in the first
month, if you factor those

00:07:23.420 --> 00:07:26.390
two things out, you're going
to just be left with a 1.

00:07:26.390 --> 00:07:29.780
In the second month if you
factor out the 50 and the 0.47,

00:07:29.780 --> 00:07:33.470
you're left with 1.47.

00:07:33.470 --> 00:07:38.780
In the third month, if you
factor that and that out,

00:07:38.780 --> 00:07:41.596
you're left with 1.47 squared.

00:07:41.596 --> 00:07:43.720
And we're going to go all
the way to the nth month.

00:07:43.720 --> 00:07:45.345
If you factor out
that and that, you're

00:07:45.345 --> 00:07:50.556
left with 1.47 to the
n minus 1th power.

00:07:50.556 --> 00:07:52.430
So which of those
expressions look like that?

00:07:52.430 --> 00:07:57.160
Well this is exactly the second
expression right over here.

00:07:57.160 --> 00:08:00.930
This is exactly what
we came up with.

00:08:00.930 --> 00:08:02.810
And we're done.