More examples of addition and subtraction of polynomials

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Kind: captions
Language: en

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In this video I want to
introduce you to the idea of a

00:00:03.430 --> 00:00:04.680
polynomial.

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It might sound like a really
fancy word, but really all it

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is is an expression that has a
bunch of variable or constant

00:00:14.580 --> 00:00:17.390
terms in them that are raised
to non-zero exponents.

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So that also probably
sounds complicated.

00:00:19.510 --> 00:00:21.330
So let me show you an example.

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If I were to give you x squared
plus 1, this is a

00:00:25.420 --> 00:00:26.260
polynomial.

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This is, in fact, a binomial
because it has two terms. The

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term polynomial is
more general.

00:00:32.159 --> 00:00:35.350
It's essentially saying you
have many terms. Poly

00:00:35.350 --> 00:00:36.300
tends to mean many.

00:00:36.300 --> 00:00:37.490
This is a binomial.

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If I were to say 4x to the third
minus 2 squared plus 7.

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This is a trinomial.

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I have three terms here.

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Let me give you just a more
concrete sense of what is and

00:00:54.330 --> 00:00:56.140
is not a polynomial.

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For example, if I were to have
x to the negative 1/2 plus 1,

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this is not a polynomial.

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That doesn't mean that you won't
ever see it while you're

00:01:08.720 --> 00:01:11.240
doing algebra or mathematics.

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But we just wouldn't call this
a polynomial because it has a

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negative and a fractional
exponent in it.

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Or if I were to give you the
expression y times the square

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root of y minus y squared.

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Once again, this is not a
polynomial, because it has a

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square root in it, which is
essentially raising something

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to the 1/2 power.

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So all of the exponents on our
variables are going to have to

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be non-negatives.

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Once again, neither of these
are polynomials.

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Now, when we're dealing with
polynomials, we're going to

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have some terminology.

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And you may or may not already
be familiar with it, so I'll

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expose it to you right now.

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The first terminology is the
degree of the polynomial.

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And essentially, that's the
highest exponent that we have

00:02:08.889 --> 00:02:09.720
in the polynomial.

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So for example, that polynomial
right there is a

00:02:13.150 --> 00:02:17.910
third degree polynomial.

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Now why is that?

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No need to keep writing it.

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Why is that a third
degree polynomial?

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Because the highest exponent
that we have in there is the x

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to the third term.

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So that's where we get it's
a third degree polynomial.

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This right here is a second
degree polynomial.

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And this is the second
degree term.

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Now a couple of other
terminologies, or words, that

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we need to know regarding
polynomials, are the constant

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versus the variable terms. And
I think you already know,

00:02:54.160 --> 00:02:56.120
these are variable
terms right here.

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This is a constant term.

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That right there is
a constant term.

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And then one last part to
dissect the polynomial

00:03:05.390 --> 00:03:08.670
properly is to understand the
coefficients of a polynomial.

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So let me write a fifth degree
polynomial here.

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And I'm going to write it in
maybe a non-conventional form

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right here.

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I'm going to not
do it in order.

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So let's just say it's x squared
minus 5x plus 7x to

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the fifth minus 5.

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So, once again, this is a
fifth degree polynomial.

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Why is that?

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Because the highest exponent
on a variable here is the 5

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right here.

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So this tells us this is a
fifth degree polynomial.

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And you might say, well why do
we even care about that?

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And at least, in my mind, the
reason why I care about the

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degree of a polynomial is
because when the numbers get

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large, the highest degree term
is what really dominates all

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of the other terms. It will
grow the fastest, or go

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negative the fastest, depending
on whether there's a

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positive or a negative
in front of it.

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But it's going to dominate
everything else.

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It really gives you a sense for
how quickly, or how fast

00:04:11.470 --> 00:04:16.269
the whole expression would grow
or decrease in the case

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if it has a negative
coefficient.

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Now I just used the
word coefficient.

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What does that mean?

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Coefficient.

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And I've used it before,
when we were just

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doing linear equations.

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And coefficients are just the
constant terms that are

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multiplying the variable
terms.

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So for example, the coefficient
on this term right

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here is negative 5.

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You have to remember we have
a minus 5, so we consider

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negative 5 to be the
whole coefficient.

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The coefficient on
this term is a 7.

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There's no coefficient here;
it's just a constant term of

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negative 5.

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And then the coefficient on
the x squared term is 1.

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The coefficient is 1.

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It's implicit.

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You're assuming it's
1 times x squared.

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Now the last thing I want to
introduce you to is just the

00:05:05.180 --> 00:05:07.190
idea of the standard form
of a polynomial.

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Now none of this is going to
help you solve a polynomial

00:05:14.710 --> 00:05:17.690
just yet, but when we talk about
solving polynomials, I

00:05:17.690 --> 00:05:20.520
might use some of this
terminology, or your teacher

00:05:20.520 --> 00:05:21.890
might use some of this
terminology.

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So it's good to know what
we're talking about.

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The standard form of a
polynomial, essentially just

00:05:28.640 --> 00:05:31.770
list the terms in
order of degree.

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So this is in a non-standard
form.

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If I were to list this
polynomial in standard form, I

00:05:37.510 --> 00:05:42.990
would put this term first. So I
would write 7x to the fifth,

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then what's the next
smallest degree?

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Well, they have this
x squared term.

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I don't have an x to
the fourth or an x

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to the third here.

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So that'll be plus 1--
well I don't have to

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write 1-- plus x squared.

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And then I have this
term, minus 5x.

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And then I have this last term
right here, minus 5.

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This is the standard form of the
polynomial where you have

00:06:07.430 --> 00:06:11.470
it in descending order
of degree.

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Now let's do a couple of
operations with polynomials.

00:06:15.280 --> 00:06:18.510
And this is going to be a super
useful toolkit later on

00:06:18.510 --> 00:06:21.730
in your algebraic, or really in
your mathematical careers.

00:06:21.730 --> 00:06:24.180
So let's just simplify a
bunch of polynomials.

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And we've kind of touched on
this in previous videos.

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But I think this will give you
a better sense, especially

00:06:29.980 --> 00:06:32.970
when we have these higher
degree terms over here.

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So let's say I wanted to
add negative 2x squared

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plus 4x minus 12.

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And I'm going to add that
to 7x plus x squared.

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Now the important thing to
remember when you simplify

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these polynomials is that you're
going to add the terms

00:06:53.430 --> 00:06:56.830
of the same variable
of like degree.

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I'll do another example in a
second where I have multiple

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variables getting involved
in the situation.

00:07:03.750 --> 00:07:05.620
But anyway, I have these
parentheses here, but they

00:07:05.620 --> 00:07:08.170
really aren't doing anything.

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If I had a subtraction sign
here, I would have to

00:07:10.220 --> 00:07:12.940
distribute the subtraction,
but I don't.

00:07:12.940 --> 00:07:16.270
So I really could just write
this as minus 2x squared, plus

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4x, minus 12, plus 7x,
plus x squared.

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And now let's simplify it.

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So let's add the terms
of like degree.

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And when I say like degree,
it has to also

00:07:28.000 --> 00:07:29.120
have the same variable.

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But in this example, we only
have the variable x.

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So let's add.

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Let's see, I have this x squared
term, and I've that x

00:07:37.090 --> 00:07:39.020
squared term, so I can
add them together.

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So I have minus 2x squared--
let me just write them

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together first --minus 2x
squared plus x squared.

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And then let me get the
x terms, so 4x and 7x.

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So this is plus 4x plus 7x.

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And then finally, I just
have this constant term

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right here, minus 12.

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And if I have negative 2 of
something, and I add 1 of

00:08:03.710 --> 00:08:05.210
something to that,
what do I have?

00:08:05.210 --> 00:08:09.750
Negative 2 plus 1 is negative
1x squared.

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I could just write negative
x squared.

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But I just want to show you that
I'm just adding negative

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2 to 1 there.

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Then I have 4x plus 7x is 11x.

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And then I finally have my
constant term, minus 12.

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And I end up with a three
term, second degree

00:08:29.390 --> 00:08:30.370
polynomial.

00:08:30.370 --> 00:08:33.740
The leading coefficient here,
the coefficient on the highest

00:08:33.740 --> 00:08:36.990
degree term in standard form--
it's already in standard form

00:08:36.990 --> 00:08:38.320
--is negative 1.

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The coefficient here is 11.

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The constant term
is negative 12.

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Let's do another one
of these examples.

00:08:43.870 --> 00:08:46.060
I think you're getting
the general idea.

00:08:46.060 --> 00:08:50.840
Now let me do a complicated
example.

00:08:50.840 --> 00:09:00.450
So let's say I have 2a squared
b, minus 3ab squared, plus 5a

00:09:00.450 --> 00:09:09.880
squared b squared, minus 2a
squared b squared, plus 4a

00:09:09.880 --> 00:09:14.110
squared b, minus 5b squared.

00:09:14.110 --> 00:09:18.290
So here, I have a minus sign,
I have multiple variables.

00:09:18.290 --> 00:09:20.990
But let's just go through
this step by step.

00:09:20.990 --> 00:09:22.220
So the first thing
you want to do is

00:09:22.220 --> 00:09:23.610
distribute this minus sign.

00:09:23.610 --> 00:09:28.470
So this first part we can just
write as 2a squared b, minus

00:09:28.470 --> 00:09:34.760
3ab squared, plus 5a
squared b squared.

00:09:34.760 --> 00:09:37.180
And then we want to distribute
this minus sign, or multiply

00:09:37.180 --> 00:09:39.760
all of these terms by negative
1 because we have

00:09:39.760 --> 00:09:41.040
the minus out here.

00:09:41.040 --> 00:09:48.150
So minus 2a squared b squared
minus 4a squared b.

00:09:48.150 --> 00:09:53.300
And a negative times a negative
is plus 5b squared.

00:09:53.300 --> 00:09:57.690
And now we want to essentially
add like terms. So I have this

00:09:57.690 --> 00:10:00.050
2a squared b squared term.

00:10:00.050 --> 00:10:02.880
So do I have any other terms
that have an a squared b

00:10:02.880 --> 00:10:04.950
squared in them?

00:10:04.950 --> 00:10:06.680
Or sorry, a squared b.

00:10:06.680 --> 00:10:07.750
I have to be very
careful here.

00:10:07.750 --> 00:10:10.860
Well, that's ab squared, no.
a squared b squared.

00:10:10.860 --> 00:10:12.360
Oh!

00:10:12.360 --> 00:10:16.580
Here I have an a squared b.

00:10:16.580 --> 00:10:17.860
So let me write those
two down.

00:10:17.860 --> 00:10:24.240
So I have 2a squared b
minus 4a squared b.

00:10:24.240 --> 00:10:26.660
That's those two terms
right there.

00:10:26.660 --> 00:10:28.940
Let me go to orange.

00:10:28.940 --> 00:10:33.280
So here I have an
ab squared term.

00:10:33.280 --> 00:10:37.480
Now do I have any other
ab squared terms here?

00:10:40.630 --> 00:10:42.550
No, no other ab squared,
so I'll just write

00:10:42.550 --> 00:10:46.740
it: minus 3 ab squared.

00:10:46.740 --> 00:10:50.580
And then let's see, I have an a
squared b squared term here.

00:10:50.580 --> 00:10:51.520
Do I have any other ones?

00:10:51.520 --> 00:10:53.390
Well, yeah sure, the
next term is.

00:10:53.390 --> 00:10:55.280
That's an a squared b
squared term, so let

00:10:55.280 --> 00:10:56.500
me just write that.

00:10:56.500 --> 00:11:01.810
Plus 5a squared b
squared minus 2a

00:11:01.810 --> 00:11:04.630
squared b squared, right?

00:11:04.630 --> 00:11:05.860
I just wrote those two.

00:11:05.860 --> 00:11:09.120
And then finally I have that
last b squared term there,

00:11:09.120 --> 00:11:12.770
plus 5b squared.

00:11:12.770 --> 00:11:14.390
Now I can add them.

00:11:14.390 --> 00:11:19.800
So this first group right here
in this purplish color, 2 of

00:11:19.800 --> 00:11:23.230
something minus 4 of something
is going to be negative 2 of

00:11:23.230 --> 00:11:23.770
that something.

00:11:23.770 --> 00:11:28.130
So it's going to be negative
2a squared b.

00:11:28.130 --> 00:11:31.450
And then this term right here,
it's not going to add to

00:11:31.450 --> 00:11:34.020
anything, 3ab squared.

00:11:34.020 --> 00:11:36.170
And then we can add these two
terms. If I have 5 of

00:11:36.170 --> 00:11:38.870
something minus 2 of something,
I'm going to have 3

00:11:38.870 --> 00:11:40.200
of that something.

00:11:40.200 --> 00:11:43.880
Plus 3a squared b squared.

00:11:43.880 --> 00:11:51.470
And then finally I have that
last term, plus 5b squared.

00:11:51.470 --> 00:11:52.000
We're done.

00:11:52.000 --> 00:11:55.170
We've simplified this
polynomial.

00:11:55.170 --> 00:11:58.990
Here, putting it in standard
form, you can think of it in

00:11:58.990 --> 00:12:00.050
different ways.

00:12:00.050 --> 00:12:02.230
The way I'd like to think of
it is maybe the combined

00:12:02.230 --> 00:12:03.290
degree of the term.

00:12:03.290 --> 00:12:06.290
Maybe we could put this one
first, but this is really

00:12:06.290 --> 00:12:08.460
according to your taste.

00:12:08.460 --> 00:12:13.030
So this is 3a squared
b squared.

00:12:13.030 --> 00:12:15.360
And then you could pick whether
you want to put the a

00:12:15.360 --> 00:12:20.680
squared b or the ab squared
terms first. 2a squared b.

00:12:20.680 --> 00:12:25.400
And then you have the
minus 3ab squared.

00:12:25.400 --> 00:12:28.320
And then we have just the
b squared term there.

00:12:28.320 --> 00:12:31.180
Plus 5b squared.

00:12:31.180 --> 00:12:32.440
And we're done.

00:12:32.440 --> 00:12:35.480
We've simplified this
polynomial.

00:12:35.480 --> 00:12:39.210
Now what I want to do next is
do a couple of examples of

00:12:39.210 --> 00:12:40.860
constructing a polynomial.

00:12:40.860 --> 00:12:43.610
And really, the idea is to give
you an appreciation for

00:12:43.610 --> 00:12:46.880
why polynomials are useful,
abstract representations.

00:12:46.880 --> 00:12:49.570
We're going to be using it all
the time, not only in algebra,

00:12:49.570 --> 00:12:51.460
but later in calculus, and
pretty much in everything.

00:12:51.460 --> 00:12:53.470
So they're really good things
to get familiar with.

00:12:53.470 --> 00:12:57.710
But what I want to do in these
four examples is represent the

00:12:57.710 --> 00:13:00.360
area of each of these figures
with a polynomial.

00:13:00.360 --> 00:13:04.690
And I'll try to match the colors
as closely as I can.

00:13:04.690 --> 00:13:06.520
So over here, what's the area?

00:13:06.520 --> 00:13:09.420
Well, this blue part right
here, the area

00:13:09.420 --> 00:13:11.315
there is x times y.

00:13:14.080 --> 00:13:15.360
And then what's the area here?

00:13:15.360 --> 00:13:17.670
It's going to be x times z.

00:13:17.670 --> 00:13:22.130
So plus x times z.

00:13:22.130 --> 00:13:22.930
But we have two of them!

00:13:22.930 --> 00:13:24.370
We have one x times z,
and then we have

00:13:24.370 --> 00:13:25.900
another x times z.

00:13:25.900 --> 00:13:27.710
So I could just add
an x times z here.

00:13:27.710 --> 00:13:31.390
Or I could just write, say,
plus 2 times x times z.

00:13:31.390 --> 00:13:36.350
And here we have a polynomial
that represents the area of

00:13:36.350 --> 00:13:38.210
this figure right there.

00:13:38.210 --> 00:13:40.470
Now let's do this next one.

00:13:40.470 --> 00:13:42.460
What's the area here?

00:13:42.460 --> 00:13:44.870
Well I have an a times a b.

00:13:44.870 --> 00:13:46.430
ab.

00:13:46.430 --> 00:13:50.140
This looks like an a times
a b again, plus ab.

00:13:50.140 --> 00:13:56.390
That looks like an ab
again, plus ab.

00:14:00.450 --> 00:14:02.090
I think they've drawn
it actually,

00:14:02.090 --> 00:14:05.460
a little bit strange.

00:14:05.460 --> 00:14:09.560
Well, I'm going to ignore
this c right there.

00:14:09.560 --> 00:14:13.740
Maybe they're telling us that
this right here is c.

00:14:13.740 --> 00:14:16.230
Because that's the information
we would need.

00:14:16.230 --> 00:14:18.860
Maybe they're telling us that
this base right there, that

00:14:18.860 --> 00:14:21.240
this right here, is c.

00:14:21.240 --> 00:14:22.850
Because that would help us.

00:14:22.850 --> 00:14:25.580
But if we assume that this is
another ab here, which I'll

00:14:25.580 --> 00:14:27.890
assume for this purpose
of this video.

00:14:27.890 --> 00:14:30.380
And then we have that last ab.

00:14:30.380 --> 00:14:33.610
And then we have this
one a times c.

00:14:36.770 --> 00:14:39.550
This is the area
of this figure.

00:14:39.550 --> 00:14:41.580
And obviously we can add
these four terms.

00:14:41.580 --> 00:14:46.780
This is 4ab and then
we have plus ac.

00:14:46.780 --> 00:14:49.400
And I made the assumption that
this was a bit of a typo, that

00:14:49.400 --> 00:14:52.310
that c where they were actually
telling us the width

00:14:52.310 --> 00:14:54.240
of this little square
over here.

00:14:54.240 --> 00:14:56.210
We don't know if it's a square,
that's only if a and c

00:14:56.210 --> 00:14:57.600
are the same.

00:14:57.600 --> 00:15:00.180
Now let's do this one.

00:15:00.180 --> 00:15:03.190
So how do we figure out the
area of the pink area?

00:15:03.190 --> 00:15:06.010
Well we could take the area
the whole rectangle, which

00:15:06.010 --> 00:15:10.730
would be 2xy, and then we could
subtract out the area of

00:15:10.730 --> 00:15:12.240
these squares.

00:15:12.240 --> 00:15:18.060
So each square has an area of
x times x, or x squared.

00:15:18.060 --> 00:15:19.810
And we have two of
these squares, so

00:15:19.810 --> 00:15:23.190
it's minus 2x squared.

00:15:23.190 --> 00:15:27.050
And then finally let's do
this one over here.

00:15:27.050 --> 00:15:30.430
So that looks like a dividing
line right there.

00:15:30.430 --> 00:15:34.250
So the area of this point, of
this area right there, is a

00:15:34.250 --> 00:15:37.130
times b, so it's ab.

00:15:37.130 --> 00:15:43.680
And then the area over here
looks like it will also be ab.

00:15:43.680 --> 00:15:45.320
So plus ab.

00:15:45.320 --> 00:15:50.950
And the area over
here is also ab.

00:15:50.950 --> 00:15:54.170
So the area here is 3ab.

00:15:54.170 --> 00:15:57.700
Anyway, hopefully that gets
us pretty warmed up with

00:15:57.700 --> 00:15:58.950
polynomials.


Multiplying binomials and polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:03.590
In this video I'm just going
to multiply a ton of

00:00:03.590 --> 00:00:05.500
polynomials, and hopefully
that'll give you enough

00:00:05.500 --> 00:00:08.630
exposure to feel confident when
you have to multiply any

00:00:08.630 --> 00:00:09.450
for yourselves.

00:00:09.450 --> 00:00:11.810
Let's start with a fairly
simple problem.

00:00:11.810 --> 00:00:18.810
Let's say we just want to
multiply 2x times 4x minus 5.

00:00:18.810 --> 00:00:20.440
Well, we just straight
up use the

00:00:20.440 --> 00:00:21.680
distributive property here.

00:00:21.680 --> 00:00:24.390
And really, when we do all
of these polynomial

00:00:24.390 --> 00:00:27.140
multiplications, all we're
doing is the distributive

00:00:27.140 --> 00:00:28.990
property repeatedly.

00:00:28.990 --> 00:00:30.710
But let's just do the
distributive property here.

00:00:30.710 --> 00:00:39.240
This is 2x times 4x, plus
2x times negative 5.

00:00:39.240 --> 00:00:41.700
Or we could say negative
5 times 2x.

00:00:41.700 --> 00:00:45.520
So you'd say, minus
5 times 2x.

00:00:45.520 --> 00:00:47.480
All I did is distribute
the 2x.

00:00:47.480 --> 00:00:50.740
This first term is going to be
equal to-- we can multiply the

00:00:50.740 --> 00:00:51.440
coefficients.

00:00:51.440 --> 00:00:56.560
Remember, 2x times 4x is the
same thing as-- you can

00:00:56.560 --> 00:00:58.530
rearrange the order
of multiplication.

00:00:58.530 --> 00:01:03.170
This is the same thing as 2
times 4, times x times x.

00:01:03.170 --> 00:01:07.020
Which is the same thing
as 8 times x squared.

00:01:07.020 --> 00:01:10.880
Remember, x to the 1, times x
to the 1, add the exponents.

00:01:10.880 --> 00:01:13.380
I mean, you know x times
x is x squared.

00:01:13.380 --> 00:01:17.070
So this first term is going
to be 8x squared.

00:01:17.070 --> 00:01:22.520
And the second term, negative
5 times 2 is negative 10x.

00:01:22.520 --> 00:01:23.720
Not too bad.

00:01:23.720 --> 00:01:26.620
Let's do a slightly
more involved one.

00:01:26.620 --> 00:01:34.140
Let's say we had 9x to the third
power, times 3x squared,

00:01:34.140 --> 00:01:38.570
minus 2x, plus 7.

00:01:38.570 --> 00:01:41.720
So once again, we're just going
to do the distributive

00:01:41.720 --> 00:01:42.810
property here.

00:01:42.810 --> 00:01:46.260
So we're going to multiply the
9x to the third times each of

00:01:46.260 --> 00:01:47.650
these terms.

00:01:47.650 --> 00:01:52.100
So 9x to the third
times 3x squared.

00:01:52.100 --> 00:01:53.750
I'll write it out this time.

00:01:53.750 --> 00:01:55.530
In the next few, we'll
start doing it a

00:01:55.530 --> 00:01:56.420
little bit in our heads.

00:01:56.420 --> 00:02:01.230
So this is going to be 9x to
the third times 3x squared.

00:02:01.230 --> 00:02:04.710
And then we're going to have
plus-- let me write it this

00:02:04.710 --> 00:02:13.960
way-- minus 2x times 9x to the
third, and then plus 7 times

00:02:13.960 --> 00:02:15.400
9x to the third.

00:02:15.400 --> 00:02:17.606
So sometimes I wrote the 9x to
the third first, sometimes we

00:02:17.606 --> 00:02:18.610
wrote it later because
I wanted this

00:02:18.610 --> 00:02:19.400
negative sign here.

00:02:19.400 --> 00:02:21.770
But it doesn't make a difference
on the order that

00:02:21.770 --> 00:02:22.980
you're multiplying.

00:02:22.980 --> 00:02:25.730
So this first term here
is going to be what?

00:02:25.730 --> 00:02:30.720
9 times 3 is 27 times x to
the-- we can add the

00:02:30.720 --> 00:02:33.190
exponents, we learned that in
our exponent properties.

00:02:33.190 --> 00:02:40.670
This is x to the fifth power,
minus 2 times 9 is 18x to

00:02:40.670 --> 00:02:43.770
the-- we have x to the
1, x to the third-- x

00:02:43.770 --> 00:02:45.970
to the fourth power.

00:02:45.970 --> 00:02:51.160
Plus 7 times 9 is 63x
to the third.

00:02:51.160 --> 00:02:54.160
So we end up with this nice
little fifth degree

00:02:54.160 --> 00:02:55.920
polynomial.

00:02:55.920 --> 00:03:01.640
Now let's do one where we are
multiplying two binomials.

00:03:01.640 --> 00:03:03.400
And I'll show you what
I mean in a second.

00:03:03.400 --> 00:03:05.680
This you're going to see
very, very, very

00:03:05.680 --> 00:03:07.510
frequently in algebra.

00:03:07.510 --> 00:03:13.215
So let's say you have x minus
3, times x plus 2.

00:03:13.215 --> 00:03:16.920
And I actually want to show you
that all we're doing here

00:03:16.920 --> 00:03:18.990
is the distributive property.

00:03:18.990 --> 00:03:25.510
So let me write it like
this: times x plus 2.

00:03:25.510 --> 00:03:27.940
So let's just pretend that this
is one big number here.

00:03:27.940 --> 00:03:28.510
And it is.

00:03:28.510 --> 00:03:31.100
You know, if you had x's, this
would be some number here.

00:03:31.100 --> 00:03:34.770
So let's just distribute this
onto each of these variables.

00:03:34.770 --> 00:03:44.790
So this is going to be x minus
3, times that green x, plus x

00:03:44.790 --> 00:03:50.760
minus 3, times that green 2.

00:03:50.760 --> 00:03:53.530
All we did is distribute
the x minus 3.

00:03:53.530 --> 00:03:55.980
This is just the distributive
property.

00:03:55.980 --> 00:04:02.920
Remember, if I had a times
x plus 2, what would

00:04:02.920 --> 00:04:03.750
this be equal to?

00:04:03.750 --> 00:04:19.810
This would be equal to a
times x plus a times 2.

00:04:19.810 --> 00:04:22.620
So over here, you can see when x
minus 3 is the same thing as

00:04:22.620 --> 00:04:24.220
a, we're just distributing it.

00:04:24.220 --> 00:04:26.910
And now we would do the
distributive property again.

00:04:26.910 --> 00:04:29.960
In this case, we're distributing
the x now onto

00:04:29.960 --> 00:04:31.050
the x minus 3.

00:04:31.050 --> 00:04:33.850
We're going to distribute the
2 onto the x minus 3.

00:04:33.850 --> 00:04:35.760
You might be used to seeing the
x on the other side, but

00:04:35.760 --> 00:04:37.730
either way, we're just
multiplying it.

00:04:37.730 --> 00:04:40.840
So this is going to be--
I'll stay color coded.

00:04:40.840 --> 00:04:56.390
This is going to be x times x,
minus 3 times x, plus x times

00:04:56.390 --> 00:04:58.870
2-- I'm going through great
pains to keep it

00:04:58.870 --> 00:05:00.030
color coded for you.

00:05:00.030 --> 00:05:04.640
I think it's helping--
minus 3 times 2.

00:05:04.640 --> 00:05:07.640
All I did is distribute the
x and distribute the 2.

00:05:07.640 --> 00:05:09.450
And soon you're going
to get used to this.

00:05:09.450 --> 00:05:10.480
We can do it in one step.

00:05:10.480 --> 00:05:13.740
You're actually multiplying
every term in this one by

00:05:13.740 --> 00:05:16.200
every term in that one, and
we'll figure out faster ways

00:05:16.200 --> 00:05:16.940
to do it in the future.

00:05:16.940 --> 00:05:18.960
But I really want to show
you the idea here.

00:05:18.960 --> 00:05:20.590
So what's this going to equal?

00:05:20.590 --> 00:05:24.050
This is going to equal
x squared.

00:05:24.050 --> 00:05:27.040
This right here is going
to be minus 3x.

00:05:27.040 --> 00:05:29.800
This is going to be plus 2x.

00:05:29.800 --> 00:05:33.720
And then this right here
is going to be minus 6.

00:05:33.720 --> 00:05:36.950
And so this is going to be x
squared minus 3 of something,

00:05:36.950 --> 00:05:39.940
plus 2 of something, that's
minus 1 of that something.

00:05:39.940 --> 00:05:42.860
Minus x, minus 6.

00:05:42.860 --> 00:05:44.510
We've multiplied those two.

00:05:44.510 --> 00:05:46.630
Now before we move on and do
another problem, I want to

00:05:46.630 --> 00:05:49.450
show you that you can kind of do
this in your head as well.

00:05:49.450 --> 00:05:51.000
You don't have to go through
all of these steps.

00:05:51.000 --> 00:05:53.670
I just want to show you really
that this is just the

00:05:53.670 --> 00:05:55.270
distributive property.

00:05:55.270 --> 00:06:00.980
The fast way of doing it, if
you had x minus 3, times x

00:06:00.980 --> 00:06:03.880
plus 2, you literally just want
to multiply every term

00:06:03.880 --> 00:06:05.240
here times each of
these terms.

00:06:05.240 --> 00:06:07.510
So you'd say, this x
times that x, so

00:06:07.510 --> 00:06:09.130
you'd have x squared.

00:06:09.130 --> 00:06:13.000
Then you'd have this x times
that 2, so plus 2x.

00:06:13.000 --> 00:06:17.250
Then you'd have this minus
3 times that x, minus 3x.

00:06:17.250 --> 00:06:20.070
And then you have the minus 3,
or the negative 3, times 2,

00:06:20.070 --> 00:06:21.750
which is negative 6.

00:06:21.750 --> 00:06:24.930
And so when you simplify, once
again you get x squared

00:06:24.930 --> 00:06:27.140
minus x minus 6.

00:06:27.140 --> 00:06:28.680
And it takes a little
bit of practice to

00:06:28.680 --> 00:06:30.680
really get used to it.

00:06:30.680 --> 00:06:32.910
Now the next thing I want to
do-- and the principal is

00:06:32.910 --> 00:06:36.160
really the exact same way-- but
I'm going to multiply a

00:06:36.160 --> 00:06:42.290
binomial times a trinomial,
which

00:06:42.290 --> 00:06:43.720
many people find daunting.

00:06:43.720 --> 00:06:46.840
But we're going to see,
if you just stay calm,

00:06:46.840 --> 00:06:48.010
it's not too bad.

00:06:48.010 --> 00:06:56.000
3x plus 2, times 9x squared,
minus 6x plus 4.

00:06:56.000 --> 00:06:58.910
Now you could do it the exact
same way that we did the

00:06:58.910 --> 00:07:00.160
previous video.

00:07:00.160 --> 00:07:03.720
We could literally take this 3x
plus 2, distribute it onto

00:07:03.720 --> 00:07:06.860
each of these three terms,
multiply 3x plus 2 times each

00:07:06.860 --> 00:07:08.860
of these terms, and then you're
going to distribute

00:07:08.860 --> 00:07:10.920
each of those terms
into 3x plus 2.

00:07:10.920 --> 00:07:13.460
It would take a long time and in
reality, you'll never do it

00:07:13.460 --> 00:07:14.090
quite that way.

00:07:14.090 --> 00:07:17.930
But you will get the same answer
we're going to get.

00:07:17.930 --> 00:07:21.130
When you have larger
polynomials, the easiest way I

00:07:21.130 --> 00:07:23.450
can think of to multiply,
is kind of how you

00:07:23.450 --> 00:07:25.730
multiply long numbers.

00:07:25.730 --> 00:07:27.300
So we'll write it like this.

00:07:27.300 --> 00:07:32.110
9x squared, minus 6, plus 4.

00:07:32.110 --> 00:07:37.010
And we're going to multiply
that times 3x plus 2.

00:07:37.010 --> 00:07:40.670
And what I imagine is, when you
multiply regular numbers,

00:07:40.670 --> 00:07:42.930
you have your ones' place,
your tens' place, your

00:07:42.930 --> 00:07:43.760
hundreds' place.

00:07:43.760 --> 00:07:46.590
Here, you're going to have your
constants' place, your

00:07:46.590 --> 00:07:49.090
first degree place, your second
degree place, your

00:07:49.090 --> 00:07:50.650
third degree place,
if there is one.

00:07:50.650 --> 00:07:52.300
And actually there will
be in this video.

00:07:52.300 --> 00:07:54.910
So you just have to put things
in their proper place.

00:07:54.910 --> 00:07:56.270
So let's do that.

00:07:56.270 --> 00:07:58.250
So you start here, you multiply
almost exactly like

00:07:58.250 --> 00:08:00.480
you would do traditional
multiplication.

00:08:00.480 --> 00:08:03.370
2 times 4 is 8.

00:08:03.370 --> 00:08:06.440
It goes into the ones', or
the constants' place.

00:08:12.520 --> 00:08:20.640
2 times negative 6x
is negative 12x.

00:08:20.640 --> 00:08:21.580
And we'll put a plus there.

00:08:21.580 --> 00:08:22.390
That was a plus 8.

00:08:22.390 --> 00:08:28.060
2 times 9x squared is 18x
squared, so we'll put that in

00:08:28.060 --> 00:08:30.360
the x squared place.

00:08:30.360 --> 00:08:32.000
Now let's do the 3x part.

00:08:32.000 --> 00:08:34.710
I'll do that in magenta, so you
see how it's different.

00:08:34.710 --> 00:08:43.480
3x times 4 is 12x,
positive 12x.

00:08:43.480 --> 00:08:48.940
3x times negative 6x,
what is that?

00:08:48.940 --> 00:08:51.190
The x times the x is
x squared, so it's

00:08:51.190 --> 00:08:53.780
going to go over here.

00:08:53.780 --> 00:08:59.330
And 3 times negative
6 is negative 18.

00:08:59.330 --> 00:09:03.330
And then finally 3x times 9x
squared, the x times the x

00:09:03.330 --> 00:09:07.150
squared is x to the
third power.

00:09:07.150 --> 00:09:09.500
3 times 9 is 27.

00:09:09.500 --> 00:09:11.670
I wrote it in the
x third place.

00:09:11.670 --> 00:09:13.620
And once again, you just
want to add the like

00:09:13.620 --> 00:09:15.230
terms. So you get 8.

00:09:15.230 --> 00:09:17.460
There's no other constant
terms, so it's just 8.

00:09:17.460 --> 00:09:20.670
Negative 12x plus 12x,
these cancel out.

00:09:20.670 --> 00:09:24.650
18x squared minus 18x squared
cancel out, so we're just left

00:09:24.650 --> 00:09:27.620
over here with 27x
to the third.

00:09:27.620 --> 00:09:33.940
So this is equal to 27x
to the third plus 8.

00:09:33.940 --> 00:09:35.110
And we are done.

00:09:35.110 --> 00:09:37.840
And you can use this technique
to multiply a trinomial times

00:09:37.840 --> 00:09:40.910
a binomial, a trinomial times
a trinomial, or really, you

00:09:40.910 --> 00:09:42.290
know, you could have
five terms up here.

00:09:42.290 --> 00:09:43.930
A fifth degree times
a fifth degree.

00:09:43.930 --> 00:09:46.480
This will always work as long
as you keep things in their

00:09:46.480 --> 00:09:49.000
proper degree place.


Exponential growth and decay word problems

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:02.910
Let's do a couple of word
problems dealing with

00:00:02.910 --> 00:00:05.940
exponential growth and decay.

00:00:05.940 --> 00:00:10.930
So this first problem, suppose a
radioactive substance decays

00:00:10.930 --> 00:00:15.130
at a rate of 3.5% per hour.

00:00:15.130 --> 00:00:19.400
What percent of the substance
is left after 6 hours?

00:00:19.400 --> 00:00:23.970
So let's make a little table
here, to just imagine

00:00:23.970 --> 00:00:24.700
what's going on.

00:00:24.700 --> 00:00:26.940
And then we'll try to come
up with a formula for, in

00:00:26.940 --> 00:00:30.210
general, how much is
left after n hours.

00:00:30.210 --> 00:00:32.980
So let's say hours that
have passed by,

00:00:32.980 --> 00:00:34.240
and percentage left.

00:00:38.270 --> 00:00:41.880
So after 0 hours, what
percent is left?

00:00:41.880 --> 00:00:45.710
Well, it hasn't decayed yet,
so we have 100% left.

00:00:45.710 --> 00:00:49.570
After 1 hour, what's happened?

00:00:49.570 --> 00:00:53.200
It decays at a rate
of 3.5% per hour.

00:00:53.200 --> 00:00:56.140
So 3.5% is gone.

00:00:56.140 --> 00:01:01.220
Or another way to think
about it is 0.965.

00:01:01.220 --> 00:01:07.860
Remember, if you take 1 minus
3.5%, or if you take 100%

00:01:07.860 --> 00:01:12.390
minus 3.5%-- this is how much
we're losing every hour-- that

00:01:12.390 --> 00:01:15.360
equals 96.5%.

00:01:15.360 --> 00:01:19.360
So each hour we're going
to have 96.5% of

00:01:19.360 --> 00:01:20.720
the previous hour.

00:01:20.720 --> 00:01:28.840
So in hour 1, we're going to
have 96.5% of hour 0, or 0.965

00:01:28.840 --> 00:01:33.970
times 100, times hour 0.

00:01:33.970 --> 00:01:35.325
Now, what happens in hour 2?

00:01:37.850 --> 00:01:41.540
Well, we're going to have 96.5%
of the previous hour.

00:01:41.540 --> 00:01:46.120
We will have lost 3.5%, which
means that we have 96.5% of

00:01:46.120 --> 00:01:47.190
the previous hour.

00:01:47.190 --> 00:01:59.870
So it'll be 0.965 times this,
times 0.965 times 100.

00:01:59.870 --> 00:02:02.660
I think you see where this
is going, in general.

00:02:02.660 --> 00:02:06.510
So in the first hour, we
have 0.965 to the first

00:02:06.510 --> 00:02:07.780
power, times 100.

00:02:07.780 --> 00:02:10.759
In the zeroth hour, we have
0.965 to the zeroth power.

00:02:10.759 --> 00:02:13.730
We don't see it, but there's
a 1 there, times 100.

00:02:13.730 --> 00:02:18.110
In the second hour, 0.965 to the
second power, times 100.

00:02:18.110 --> 00:02:22.040
So in general, in the nth hour--
let me do this in a

00:02:22.040 --> 00:02:25.860
nice bold color-- in the nth
hour, we're going to have

00:02:25.860 --> 00:02:33.850
0.965 to the nth power,
times 100 left of

00:02:33.850 --> 00:02:36.130
our radioactive substance.

00:02:36.130 --> 00:02:38.100
And oftentimes you'll see
it written this way.

00:02:38.100 --> 00:02:44.380
You have your initial amount
times your common ratio, 0.965

00:02:44.380 --> 00:02:45.450
to the nth power.

00:02:45.450 --> 00:02:48.500
This is how much you're going
to have left after n hours.

00:02:48.500 --> 00:02:49.820
Well, now we can answer
the question.

00:02:49.820 --> 00:02:52.310
After 6 hours how much are
we going to have left?

00:02:52.310 --> 00:02:58.860
Well, we're going to have
100 times 0.965 to

00:02:58.860 --> 00:03:00.520
the sixth power left.

00:03:00.520 --> 00:03:04.150
And we could use a calculator
to figure out what that is.

00:03:04.150 --> 00:03:07.000
Let's use our trusty
calculator.

00:03:07.000 --> 00:03:16.880
So we have 100 times 0.965 to
the sixth power, which is

00:03:16.880 --> 00:03:21.820
equal to 80.75.

00:03:21.820 --> 00:03:23.080
This is all in percentages.

00:03:23.080 --> 00:03:27.220
So it's 80.75% of our
original substance.

00:03:32.600 --> 00:03:34.646
Let's do another one of these.

00:03:34.646 --> 00:03:41.570
So we have, Nadia owns a chain
of fast food restaurants that

00:03:41.570 --> 00:03:48.360
operated 200 stores in 1999.

00:03:48.360 --> 00:03:51.500
If the rate of increase is--
oh actually, there's a typo

00:03:51.500 --> 00:03:56.670
here, it should be 8%-- the
rate of increase is 8%

00:03:56.670 --> 00:03:58.760
annually, how many
stores does the

00:03:58.760 --> 00:04:02.450
restaurant operate in 2007?

00:04:02.450 --> 00:04:06.400
So let's think about
the same thing.

00:04:06.400 --> 00:04:13.860
So let's say years after 1999.

00:04:13.860 --> 00:04:21.019
And let's talk about how many
stores Nadia is operating, her

00:04:21.019 --> 00:04:22.820
fast food chain.

00:04:22.820 --> 00:04:29.950
So 1999 itself is 0
years after 1999.

00:04:29.950 --> 00:04:33.710
And she is operating
200 stores.

00:04:33.710 --> 00:04:38.620
Then in 2000, which is 1 year
after 1999, how many is she

00:04:38.620 --> 00:04:40.040
going to be operating?

00:04:40.040 --> 00:04:43.190
Well, she grows at the
rate of 8% annually.

00:04:43.190 --> 00:04:48.060
So she'll be operating all the
stores that she had before

00:04:48.060 --> 00:04:51.210
plus 8% of the store
she had before.

00:04:51.210 --> 00:04:55.980
So 1.08 times the number of
stores she had before.

00:04:55.980 --> 00:04:58.660
And you're going to see, the
common ratio here is 1.08.

00:04:58.660 --> 00:05:01.380
If you're growing by 8%,
that's equivalent to

00:05:01.380 --> 00:05:03.460
multiplying by 1.08.

00:05:03.460 --> 00:05:04.740
Let me make that clear.

00:05:04.740 --> 00:05:09.290
200 plus 0.08, times 200.

00:05:09.290 --> 00:05:15.040
Well, this is just 1 times
200 plus 0.08, times 200.

00:05:15.040 --> 00:05:17.700
That's 1.08 times 200.

00:05:17.700 --> 00:05:21.920
Then in 2001, what's going on?

00:05:21.920 --> 00:05:27.310
This is now 2 years after 1999,
and you're going to grow

00:05:27.310 --> 00:05:29.230
8% from this number.

00:05:29.230 --> 00:05:33.330
You're going to multiply
1.08 times that number,

00:05:33.330 --> 00:05:37.510
times 1.08 times 200.

00:05:37.510 --> 00:05:42.060
I think you get the general
gist. If, after n years after

00:05:42.060 --> 00:05:46.550
1999, it's going to be 1.08--
let me write it this way.

00:05:46.550 --> 00:05:54.660
It's going to be 200 times
1.08 to the nth power.

00:05:54.660 --> 00:05:57.200
After 2 years, 1.08 squared.

00:05:57.200 --> 00:05:59.370
1 year, 1.08 to the
first power.

00:05:59.370 --> 00:06:02.960
0 years, this is the same thing
as a 1 times 200, which

00:06:02.960 --> 00:06:05.870
is 1.08 to the zeroth power.

00:06:05.870 --> 00:06:08.620
So they're asking us, how many
stores does the restaurant

00:06:08.620 --> 00:06:10.550
operate in 2007?

00:06:10.550 --> 00:06:17.320
Well, 2007 is 8 years
after 1999.

00:06:17.320 --> 00:06:21.090
So here n is equal to 8.

00:06:21.090 --> 00:06:24.110
So let us substitute
n is equal to 8.

00:06:24.110 --> 00:06:29.180
The answer to our question will
be 200 times 1.08 to the

00:06:29.180 --> 00:06:30.410
eighth power.

00:06:30.410 --> 00:06:34.000
Let's get our calculator
out and calculate it.

00:06:34.000 --> 00:06:41.570
So we want to figure
out 200 times 1.08

00:06:41.570 --> 00:06:46.090
to the eighth power.

00:06:46.090 --> 00:06:49.840
She's going to be operating 370
restaurants, and she'll be

00:06:49.840 --> 00:06:52.250
in the process of opening
a few more.

00:06:52.250 --> 00:06:58.050
So if we round it down, she's
going to be operating 370

00:06:58.050 --> 00:06:58.900
restaurants.

00:06:58.900 --> 00:07:03.255
So 8% growth might not look like
something that's so fast

00:07:03.255 --> 00:07:04.910
or that exciting.

00:07:04.910 --> 00:07:08.430
But in under a decade, in only 8
years, she would have gotten

00:07:08.430 --> 00:07:12.390
her restaurant chain from
200 to 370 restaurants.

00:07:12.390 --> 00:07:17.020
So over 8 years, you see that
the compounding growth by 8%

00:07:17.020 --> 00:07:20.440
actually ends up being
quite dramatic.


Introduction to geometric sequences

WEBVTT
Kind: captions
Language: en

00:00:00.410 --> 00:00:03.520
In this video I want to
introduce you to the idea of a

00:00:03.520 --> 00:00:05.245
geometric sequence.

00:00:09.590 --> 00:00:12.720
And I have a ton of more
advanced videos on the topic,

00:00:12.720 --> 00:00:15.330
but it's really a good place to
start, just to understand

00:00:15.330 --> 00:00:17.380
what we're talking about
when someone tells

00:00:17.380 --> 00:00:19.450
you a geometric sequence.

00:00:19.450 --> 00:00:22.200
Now a good starting point is
just, what is a sequence?

00:00:22.200 --> 00:00:24.330
And a sequence is, you
can imagine, just a

00:00:24.330 --> 00:00:26.420
progression of numbers.

00:00:26.420 --> 00:00:29.730
So for example, and this isn't
even a geometric series, if I

00:00:29.730 --> 00:00:34.570
just said 1, 2, 3, 4, 5.

00:00:34.570 --> 00:00:37.120
This is a sequence of numbers.

00:00:37.120 --> 00:00:40.680
It's not a geometric sequence,
but it is a sequence.

00:00:40.680 --> 00:00:45.085
A geometric sequence is a
special progression, or a

00:00:45.085 --> 00:00:52.370
special sequence, of numbers,
where each successive number

00:00:52.370 --> 00:00:55.180
is a fixed multiple of
the number before it.

00:00:55.180 --> 00:00:57.080
Let me explain what
I'm saying.

00:00:57.080 --> 00:01:03.430
So let's say my first number is
2 and then I multiply 2 by

00:01:03.430 --> 00:01:05.290
the number 3.

00:01:05.290 --> 00:01:07.420
So I multiply it
by 3, I get 6.

00:01:07.420 --> 00:01:10.590
And then I multiply 6 times the
number 3, and I get 18.

00:01:10.590 --> 00:01:15.230
Then I multiply 18 times the
number 3, and I get 54.

00:01:15.230 --> 00:01:17.280
And I just keep going
that way.

00:01:17.280 --> 00:01:21.090
So I just keep multiplying
by the number 3.

00:01:21.090 --> 00:01:24.860
So I started, if we want to get
some notation here, this

00:01:24.860 --> 00:01:26.970
is my first term.

00:01:26.970 --> 00:01:29.840
We'll call it a1 for
my sequence.

00:01:29.840 --> 00:01:34.170
And each time I'm multiplying
it by a common number, and

00:01:34.170 --> 00:01:37.140
that number is often called
the common ratio.

00:01:37.140 --> 00:01:42.420
So in this case, a1 is equal to
2, and my common ratio is

00:01:42.420 --> 00:01:43.670
equal to 3.

00:01:49.760 --> 00:01:52.680
So if someone were to tell
you, hey, you've got a

00:01:52.680 --> 00:01:54.310
geometric sequence.

00:01:54.310 --> 00:01:59.220
a1 is equal to 90 and your
common ratio is equal to

00:01:59.220 --> 00:02:01.160
negative 1/3.

00:02:01.160 --> 00:02:05.570
That means that the first term
of your sequence is 90.

00:02:05.570 --> 00:02:09.770
The second term is negative
1/3 times 90.

00:02:09.770 --> 00:02:10.759
Which is what?

00:02:10.759 --> 00:02:14.160
That's negative 30, right?

00:02:14.160 --> 00:02:19.340
1/3 times 90 is 30, and then you
put the negative number.

00:02:19.340 --> 00:02:23.000
Then the next number is going
to be 1/3 times this.

00:02:23.000 --> 00:02:25.930
So negative 1/3 times this.

00:02:25.930 --> 00:02:27.790
1/3 times 30 is 10.

00:02:27.790 --> 00:02:30.790
The negatives cancel out,
so you get positive 10.

00:02:30.790 --> 00:02:33.650
Then the next number is going
to be 10 times negative 1/3,

00:02:33.650 --> 00:02:36.250
or negative 10/3.

00:02:36.250 --> 00:02:38.500
And then the next number is
going to be negative 10/3

00:02:38.500 --> 00:02:41.950
times negative 1/3 so it's going
to be positive 10/3.

00:02:41.950 --> 00:02:45.950
And you could just keep going
on with this sequence.

00:02:45.950 --> 00:02:49.040
So that's what people talk
about when they mean a

00:02:49.040 --> 00:02:51.360
geometric sequence.

00:02:51.360 --> 00:02:52.880
I want to make one little
distinction here.

00:02:52.880 --> 00:02:56.280
This always used to confuse me
because the terms are used

00:02:56.280 --> 00:02:58.940
very often in the
same context.

00:02:58.940 --> 00:02:59.980
These are sequences.

00:02:59.980 --> 00:03:01.860
These are kind of a progression
of numbers.

00:03:01.860 --> 00:03:05.330
2, then 6, then 18, 90, then
negative 30, then 10, then

00:03:05.330 --> 00:03:07.030
negative 10/3.

00:03:07.030 --> 00:03:11.150
Then, I'm sorry, this is
positive 10/9, right?

00:03:11.150 --> 00:03:14.500
Negative 1/3 times negative
10/3, negatives cancel out.

00:03:14.500 --> 00:03:14.760
Right.

00:03:14.760 --> 00:03:15.550
10/9.

00:03:15.550 --> 00:03:16.830
Don't want to make
a mistake here.

00:03:16.830 --> 00:03:18.350
These are sequences.

00:03:18.350 --> 00:03:20.725
You might also see the
word a series.

00:03:24.840 --> 00:03:31.180
And you might even see
a geometric series.

00:03:31.180 --> 00:03:34.360
A series, the most conventional
use of the word

00:03:34.360 --> 00:03:38.410
series, means a sum
of a sequence.

00:03:38.410 --> 00:03:41.850
So for example, this is
a geometric sequence.

00:03:41.850 --> 00:03:49.070
A geometric series would be 90
plus negative 30, plus 10,

00:03:49.070 --> 00:03:53.500
plus negative 10/3, plus 10/9.

00:03:53.500 --> 00:03:56.660
So a general way to view
it is that a series is

00:03:56.660 --> 00:03:59.910
the sum of a sequence.

00:03:59.910 --> 00:04:02.470
I just want to make that clear
because that used to confuse

00:04:02.470 --> 00:04:05.690
me a lot when I first learned
about these things.

00:04:05.690 --> 00:04:08.550
But anyway, let's go back to
the notion of a geometric

00:04:08.550 --> 00:04:12.240
sequence, and actually do a word
problem that deals with

00:04:12.240 --> 00:04:13.520
one of these.

00:04:13.520 --> 00:04:18.959
So they're telling us that Anne
goes bungee jumping off

00:04:18.959 --> 00:04:20.599
of a bridge above water.

00:04:30.570 --> 00:04:35.690
On the initial jump, the cord
stretches by 120 feet.

00:04:35.690 --> 00:04:39.750
So on a1, our initial
jump, the cord

00:04:39.750 --> 00:04:42.260
stretches by 120 feet.

00:04:42.260 --> 00:04:43.510
We could write it this way.

00:04:46.100 --> 00:04:51.865
We could write, jump, and then
how much the cord stretches.

00:04:56.330 --> 00:04:59.370
So on the initial jump,
on jump one, the cord

00:04:59.370 --> 00:05:02.870
stretches 120 feet.

00:05:02.870 --> 00:05:06.420
Then it says, on the next
bounce, the stretch is 60% of

00:05:06.420 --> 00:05:09.910
the original jump, and then
each additional bounce

00:05:09.910 --> 00:05:13.230
stretches the rope 60% of
the previous stretch.

00:05:13.230 --> 00:05:17.280
So here, the common ratio, where
each successive term in

00:05:17.280 --> 00:05:20.400
our sequence is going to be
60% of the previous term.

00:05:20.400 --> 00:05:24.680
Or it's going to be 0.6 times
the previous term.

00:05:24.680 --> 00:05:28.400
So on the second jump, we're
going to start 60% of that, or

00:05:28.400 --> 00:05:32.220
0.6 times 120.

00:05:32.220 --> 00:05:33.180
Which is equal to what?

00:05:33.180 --> 00:05:35.960
That's equal to 72.

00:05:35.960 --> 00:05:39.210
Then on the third jump, we're
going to stretch 0.6 of 72, or

00:05:39.210 --> 00:05:40.470
0.6 times this.

00:05:40.470 --> 00:05:49.900
So it would be 0.6 times
0.6 times 120.

00:05:49.900 --> 00:05:53.410
Notice, over here, so on the
fourth jump we're going to

00:05:53.410 --> 00:06:03.770
have 0.6 times 0.6 times
0.6 times 120.

00:06:03.770 --> 00:06:07.810
60% of this jump, so every
time we're 60% of

00:06:07.810 --> 00:06:09.030
the previous jump.

00:06:09.030 --> 00:06:12.060
So if we wanted to make a
general formula for this, just

00:06:12.060 --> 00:06:15.690
based on the way we've defined
it right here.

00:06:15.690 --> 00:06:25.960
So the stretch on the nth
jump, what would it be?

00:06:25.960 --> 00:06:39.900
So let's see, we start at 120
times 0.6 to the what?

00:06:39.900 --> 00:06:42.570
To the n minus 1.

00:06:42.570 --> 00:06:43.330
How did I get this?

00:06:43.330 --> 00:06:45.410
Let me write this a
little bit here.

00:06:45.410 --> 00:06:50.620
So this is equal to 0.6,
actually let me write the 120

00:06:50.620 --> 00:06:59.500
first. This is equal to 120
times 0.6 to the n minus 1.

00:06:59.500 --> 00:07:00.540
How did I get that?

00:07:00.540 --> 00:07:04.610
Well we're defining the first
jump as stretching 120 feet.

00:07:04.610 --> 00:07:09.880
So when you put n is equal to 1
here, you get 1 minus 1, 0.

00:07:09.880 --> 00:07:11.995
So you have 0.6 to the
0th power, and you've

00:07:11.995 --> 00:07:13.360
just got a 1 here.

00:07:13.360 --> 00:07:15.920
And that's exactly what happened
on the first jump.

00:07:15.920 --> 00:07:20.360
Then on the second jump, you put
a 2 minus 1, and notice 2

00:07:20.360 --> 00:07:22.050
minus 1 is the first
power, and we have

00:07:22.050 --> 00:07:23.900
exactly one 0.6 here.

00:07:23.900 --> 00:07:27.620
So I figured it was n minus 1
because when n is 2, we have

00:07:27.620 --> 00:07:31.560
one 0.6, when n is 3, we have
two 0.6's multiplied by

00:07:31.560 --> 00:07:32.040
themselves.

00:07:32.040 --> 00:07:35.120
When n is 4, we have 0.6
to the third power.

00:07:35.120 --> 00:07:39.510
So whatever n is, we're taking
0.6 to the n minus 1 power,

00:07:39.510 --> 00:07:43.940
and of course we're multiplying
that times 120.

00:07:43.940 --> 00:07:47.340
Now and the question they also
ask us, what will be the rope

00:07:47.340 --> 00:07:54.390
stretch on the 12th bounce?

00:07:54.390 --> 00:08:00.790
And over here I'm going
to use the calculator.

00:08:00.790 --> 00:08:04.600
and actually let me correct
this a little bit.

00:08:04.600 --> 00:08:06.550
This isn't incorrect, but
they're talking about the

00:08:06.550 --> 00:08:11.150
bounce, and we could call the
jump the zeroth bounce.

00:08:11.150 --> 00:08:13.190
Let me change that.

00:08:13.190 --> 00:08:17.150
This isn't wrong, but I think
this is where they're going

00:08:17.150 --> 00:08:18.340
with the problem.

00:08:18.340 --> 00:08:23.300
So you can view the initial
stretch as the zeroth bounce.

00:08:23.300 --> 00:08:28.050
So instead of labeling it jump,
let me label it bounce.

00:08:28.050 --> 00:08:32.049
So the initial stretch is the
zeroth bounce, then this would

00:08:32.049 --> 00:08:33.960
be the first bounce, the second

00:08:33.960 --> 00:08:35.950
bounce, the third bounce.

00:08:35.950 --> 00:08:38.960
And then our formula becomes
a lot simpler.

00:08:38.960 --> 00:08:48.150
Because if you said the stretch
on nth bounce, then

00:08:48.150 --> 00:08:57.180
the formula just becomes 0.6
to the n times 120, right?

00:08:57.180 --> 00:09:00.740
On the zeroth bounce, that was
our original stretch, you get

00:09:00.740 --> 00:09:03.660
0.6 to the 0, that's
1 times 120.

00:09:03.660 --> 00:09:09.210
On the first bounce, 0.6 to
the 1, one 0.6 right here.

00:09:09.210 --> 00:09:11.860
0.6 times the previous
stretch, or

00:09:11.860 --> 00:09:13.350
the previous bounce.

00:09:13.350 --> 00:09:16.390
So this has it in terms of
bounces, which I think is what

00:09:16.390 --> 00:09:18.690
the questioner wants us to do.

00:09:18.690 --> 00:09:22.940
So what about the 12th bounce?

00:09:22.940 --> 00:09:25.500
Using this convention
right there.

00:09:25.500 --> 00:09:29.240
So if we do the 12th bounce,
let's just get

00:09:29.240 --> 00:09:30.930
our calculator out.

00:09:30.930 --> 00:09:41.790
We're going to have 120 times
0.6 to the 12th power.

00:09:41.790 --> 00:09:44.250
And hopefully we'll get order
of operations right, because

00:09:44.250 --> 00:09:47.390
exponents take precedence over
multiplication, so it'll just

00:09:47.390 --> 00:09:50.290
take the 0.6 to the
12th power only.

00:09:50.290 --> 00:09:57.930
And so this is equal
to 0.26 feet.

00:09:57.930 --> 00:09:59.990
So after your 12th bounce,
she's going

00:09:59.990 --> 00:10:00.980
to be barely moving.

00:10:00.980 --> 00:10:04.450
She's going to be moving about
3 inches on that 12th bounce.

00:10:04.450 --> 00:10:05.760
Well, hopefully you found
this helpful.

00:10:05.760 --> 00:10:08.360
And I apologize for the slight
divergence here, but I

00:10:08.360 --> 00:10:10.420
actually think on some level
that's instructive.

00:10:10.420 --> 00:10:13.750
Because you always have to make
sure that your n matches

00:10:13.750 --> 00:10:15.740
well with what your
results are.

00:10:15.740 --> 00:10:19.080
So when I talked about
your first jump, I

00:10:19.080 --> 00:10:20.730
said, OK this is 1.

00:10:20.730 --> 00:10:23.686
And then I had 0.6 to
the zeroth power, so

00:10:23.686 --> 00:10:24.850
I did n minus 1.

00:10:24.850 --> 00:10:27.370
But then when I relabeled things
in terms of bounces,

00:10:27.370 --> 00:10:29.070
this was the zeroth bounce.

00:10:29.070 --> 00:10:31.960
This makes sense that this is
0.6 to the zeroth power.

00:10:31.960 --> 00:10:35.000
This is the first bounce,
so this would be 0.6

00:10:35.000 --> 00:10:36.500
to the first power.

00:10:36.500 --> 00:10:39.250
Second bounce, 0.6 to
the second power.

00:10:39.250 --> 00:10:42.190
It made our equation a
little bit simpler.

00:10:42.190 --> 00:10:44.730
Anyway, hopefully you found
that Interesting.


Exponential Decay Functions

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.010
In the last video on exponential
functions, we saw

00:00:03.010 --> 00:00:06.380
that functions of the form,
y is equal to some

00:00:06.380 --> 00:00:07.180
number to the x.

00:00:07.180 --> 00:00:09.050
Say y is equal to 3x,
which was what we

00:00:09.050 --> 00:00:10.410
used in the last video.

00:00:10.410 --> 00:00:14.630
We saw that these grow very,
very, very quickly.

00:00:14.630 --> 00:00:17.960
So the shape of the graph that
we got in the last video looks

00:00:17.960 --> 00:00:21.060
something like this.

00:00:21.060 --> 00:00:22.790
It gets very close to 0,
slowly pulls away from

00:00:22.790 --> 00:00:23.920
0, and then, bam!

00:00:23.920 --> 00:00:24.870
It starts to explode.

00:00:24.870 --> 00:00:27.010
This, of course, is when
x is equal to 0.

00:00:27.010 --> 00:00:29.970
So the y-intercept there
is y is equal to 1.

00:00:29.970 --> 00:00:32.990
But the important thing is it
just grows super fast. But

00:00:32.990 --> 00:00:36.090
this was the case where the
base in the exponential

00:00:36.090 --> 00:00:39.430
function is greater then 1.

00:00:39.430 --> 00:00:40.280
Here, it's 3.

00:00:40.280 --> 00:00:41.900
If it's 2, it'll still
grow quickly.

00:00:41.900 --> 00:00:44.840
If it was 10, it would
grow even faster.

00:00:44.840 --> 00:00:46.790
What I want to talk a little bit
about in this video, and

00:00:46.790 --> 00:00:49.990
really just introduce you to the
idea, is the notion of an

00:00:49.990 --> 00:00:52.320
exponential decay function.

00:00:52.320 --> 00:00:56.530
So this is an exponential
growth function.

00:00:56.530 --> 00:01:00.540
An exponential decay function
has essentially, as its base,

00:01:00.540 --> 00:01:02.530
a number less than 1.

00:01:02.530 --> 00:01:04.790
So an example of an exponential
decay function

00:01:04.790 --> 00:01:10.600
would be y is to 1/3
to the x power.

00:01:10.600 --> 00:01:17.180
This would be exponential
decay.

00:01:17.180 --> 00:01:20.620
And notice, another way I could
write this, 1/3 is the

00:01:20.620 --> 00:01:23.480
same thing as 3 to
the negative 1.

00:01:23.480 --> 00:01:26.490
So it's 3 to the negative
1 to the x power.

00:01:26.490 --> 00:01:30.560
I could also write this as 3
to the negative x power.

00:01:30.560 --> 00:01:32.940
This and this are equivalent.

00:01:32.940 --> 00:01:35.440
That's another interesting way
to think about exponential

00:01:35.440 --> 00:01:36.380
decay functions.

00:01:36.380 --> 00:01:39.410
It could either be kind of a
standard positive exponent,

00:01:39.410 --> 00:01:42.750
with a base that's less than
1, or you could have a base

00:01:42.750 --> 00:01:46.110
larger than 1, but you're going
to have a negative as

00:01:46.110 --> 00:01:46.980
the exponent.

00:01:46.980 --> 00:01:49.520
So let's plot this, just to
get the general shape.

00:01:49.520 --> 00:01:51.140
And the shape will always
be consistent.

00:01:51.140 --> 00:01:54.260
I'm going to do it for 1/3,
but it's going to have a

00:01:54.260 --> 00:01:58.010
similar shape for 1/2, or
for 1/10, or for 1/100.

00:01:58.010 --> 00:02:00.320
It'll just be scaled
up or down.

00:02:00.320 --> 00:02:02.050
So let's plot a few.

00:02:02.050 --> 00:02:06.890
Let's have some x values and
let's have some y values.

00:02:06.890 --> 00:02:10.580
So let's start with
some small values.

00:02:10.580 --> 00:02:14.380
So let's say that x is
equal to negative 3.

00:02:14.380 --> 00:02:19.200
Well, what is 1/3 to the
negative 3 power?

00:02:19.200 --> 00:02:23.740
1/3 to the negative 3, that's
the same thing as 3 to the

00:02:23.740 --> 00:02:27.440
third power, which
is equal to 81.

00:02:27.440 --> 00:02:28.310
Let's do another point.

00:02:28.310 --> 00:02:30.070
What about negative 2?

00:02:30.070 --> 00:02:32.800
1/3 to the negative 2 power.

00:02:32.800 --> 00:02:37.380
It's the same thing as 1 over--
Sorry, it's equal to 1

00:02:37.380 --> 00:02:41.470
over 1/3 squared, which the
same thing as 3 squared.

00:02:41.470 --> 00:02:43.840
You just take the inverse of
the number, get rid of the

00:02:43.840 --> 00:02:47.590
negative and that is equal
to equal to 9.

00:02:47.590 --> 00:02:49.620
Oh, I got this one
wrong up here.

00:02:49.620 --> 00:02:50.890
3 cubed isn't 81.

00:02:50.890 --> 00:02:52.140
That's 3 to the fourth.

00:02:52.140 --> 00:02:56.010
3 times 3 times 3 is 27.

00:02:56.010 --> 00:02:58.980
My brain is malfunctioning.

00:02:58.980 --> 00:03:03.040
What about when x is equal to
negative 1, then you have 1/3

00:03:03.040 --> 00:03:06.370
to the negative 1 power.

00:03:06.370 --> 00:03:08.280
which is just 3.

00:03:08.280 --> 00:03:12.030
What about when x
is equal to 0?

00:03:12.030 --> 00:03:14.860
When x is equal to 0, remember,
anything to the 0

00:03:14.860 --> 00:03:17.070
power is 1.

00:03:17.070 --> 00:03:19.360
And then let's go to a couple
of larger numbers.

00:03:19.360 --> 00:03:21.630
When x is equal to
1, what is y?

00:03:21.630 --> 00:03:24.990
1/3 to the 1 power is 1/3.

00:03:24.990 --> 00:03:26.770
Let's do one more.

00:03:26.770 --> 00:03:32.920
When x is equal to 2, we have
1/3 squared, which is 1/9.

00:03:32.920 --> 00:03:39.200
So let's plot this exponential
decay function right here.

00:03:39.200 --> 00:03:40.590
I want to draw it reasonably
neatly.

00:03:44.660 --> 00:03:46.870
And all of these values, once
again, are positive because

00:03:46.870 --> 00:03:49.260
our base was positive.

00:03:49.260 --> 00:03:52.175
So it looks like that.

00:03:55.530 --> 00:04:02.020
So let's say that this
is 10, 20, 30.

00:04:02.020 --> 00:04:07.270
Let's start at negative
3, negative 2,

00:04:07.270 --> 00:04:11.380
negative 1, 0, 1, 2.

00:04:11.380 --> 00:04:16.880
So when x was negative
3, y was 27.

00:04:16.880 --> 00:04:18.589
Right about there.

00:04:18.589 --> 00:04:21.200
When x was negative
2, y was 9.

00:04:21.200 --> 00:04:23.650
Which is right about there.

00:04:23.650 --> 00:04:27.000
When x is negative 1, y is 3.

00:04:27.000 --> 00:04:30.630
Right about there.
x is 0, y is 1.

00:04:30.630 --> 00:04:34.825
x is 1, y is 1/3, until
we decline like this.

00:04:38.080 --> 00:04:41.350
Let me draw it a little
bit neater than that.

00:04:41.350 --> 00:04:43.110
The decay functions are
harder for me to draw.

00:04:46.820 --> 00:04:49.670
We'll draw it in
reverse order.

00:04:49.670 --> 00:04:50.520
No, that's even worse.

00:04:50.520 --> 00:04:51.650
One more shot.

00:04:51.650 --> 00:04:54.760
I think you get the
general idea.

00:04:54.760 --> 00:04:56.700
That's my best shot
right there.

00:04:56.700 --> 00:04:59.030
So you see, a decay function
is kind of

00:04:59.030 --> 00:05:00.950
reversed around the y-axis.

00:05:00.950 --> 00:05:03.700
We start from a large quantity
and we decay

00:05:03.700 --> 00:05:05.430
very, very, very quickly.

00:05:05.430 --> 00:05:08.590
So in either case, exponential
functions, whether we have

00:05:08.590 --> 00:05:11.920
exponential growth or
exponential decay-- Let me

00:05:11.920 --> 00:05:14.210
make sure that you understand
this growth applies to this

00:05:14.210 --> 00:05:15.280
thing up here.

00:05:15.280 --> 00:05:17.220
This is exponential decay.

00:05:17.220 --> 00:05:19.800
And either way, in exponential
functions,

00:05:19.800 --> 00:05:22.470
things happen quickly.

00:05:22.470 --> 00:05:26.330
Now, using what we know about
exponential decay functions,

00:05:26.330 --> 00:05:30.460
let's do an actual
word problem.

00:05:30.460 --> 00:05:31.320
Let me read it here.

00:05:31.320 --> 00:05:34.545
It says, a person is infected
by a bacterial infection.

00:05:40.880 --> 00:05:44.110
And they say, when he goes to
the doctor, the population of

00:05:44.110 --> 00:05:46.200
bacteria is 2 million.

00:05:46.200 --> 00:05:55.696
So at the start, he's got 2
million little critters, 2

00:05:55.696 --> 00:05:58.260
million bacteria.

00:05:58.260 --> 00:06:01.140
The doctor prescribes an
antibiotic which kills

00:06:01.140 --> 00:06:04.700
bacteria, that reduces the
bacteria to 1/4 of its

00:06:04.700 --> 00:06:06.655
population each day.

00:06:11.460 --> 00:06:19.060
So we have the antibiotic and
it cuts down the population.

00:06:19.060 --> 00:06:33.430
So it reduces population
to 1/4 of the previous

00:06:33.430 --> 00:06:39.270
population each day.

00:06:39.270 --> 00:06:41.020
So they ask us a couple
of things.

00:06:41.020 --> 00:06:44.920
Draw the graph of the size of
the bacteria population

00:06:44.920 --> 00:06:46.840
against time in days.

00:06:46.840 --> 00:06:47.870
So let's draw this.

00:06:47.870 --> 00:06:51.960
So day, and then population.

00:06:56.870 --> 00:07:00.250
So let's say on day 0, before
you went to the doctor, your

00:07:00.250 --> 00:07:04.060
population is going
to be 2 million.

00:07:04.060 --> 00:07:06.750
Now on day 1, what happens?

00:07:06.750 --> 00:07:09.450
On day 1 you've taken their
antibiotic, right?

00:07:09.450 --> 00:07:11.540
The antibiotic is starting
right there.

00:07:11.540 --> 00:07:14.510
So it reduces it to 1/4 of
your previous population.

00:07:14.510 --> 00:07:18.090
So now your population is
going to be 1/4 times 2

00:07:18.090 --> 00:07:25.620
million, which is equal
to 500,000.

00:07:25.620 --> 00:07:27.800
Then on day 2, what happens?

00:07:27.800 --> 00:07:29.940
On day 2, it's going
to be 1/4 of this.

00:07:29.940 --> 00:07:32.660
It's going to be 1/4 of
your previous day.

00:07:32.660 --> 00:07:37.790
1/4 of 1/4 of your previous
2 million.

00:07:40.470 --> 00:07:46.450
So it's 1/4 of this 500,000
which is 125,000.

00:07:46.450 --> 00:07:49.450
I think you're starting to see
the pattern that's emerging.

00:07:49.450 --> 00:07:51.010
On day 3, what's
it going to be?

00:07:51.010 --> 00:07:53.140
It's going to be 1/4
times this thing.

00:07:53.140 --> 00:08:00.720
So it's going to be 1/4 times
this 1/4 times 1/4 times 1/4

00:08:00.720 --> 00:08:03.890
times 2 million.

00:08:03.890 --> 00:08:07.000
Which is essentially 125,000
divided by 4, but the

00:08:07.000 --> 00:08:10.890
important thing is, I want
you to see this pattern.

00:08:10.890 --> 00:08:14.730
In general, the population
at day n-- Let me write

00:08:14.730 --> 00:08:16.400
this in a new color.

00:08:16.400 --> 00:08:25.850
The population on day n is going
to be equal to what?

00:08:25.850 --> 00:08:29.600
It's going to be equal to 1/4
to the n power, right?

00:08:29.600 --> 00:08:30.730
This is day n.

00:08:30.730 --> 00:08:32.630
On the third day, it was
1/4 to the third

00:08:32.630 --> 00:08:33.700
power times 2 million.

00:08:33.700 --> 00:08:36.659
On day 2, it was 1/4 to
the second power.

00:08:36.659 --> 00:08:38.799
On day 1 it was 1/4 to
the first power.

00:08:38.799 --> 00:08:41.690
On day 0, there is actually
1/4 to the 0 power, right?

00:08:41.690 --> 00:08:42.799
That's 1.

00:08:42.799 --> 00:08:48.660
So it's 1/4 to the nth power
times our starting population,

00:08:48.660 --> 00:08:51.606
times 2 million.

00:08:54.710 --> 00:08:59.310
And if we were to graph it,
using at least just the points

00:08:59.310 --> 00:09:01.870
that we have here, we're only
going to be dealing with

00:09:01.870 --> 00:09:03.630
positive x's.

00:09:03.630 --> 00:09:07.360
So on day 0, we start
at 2 million.

00:09:07.360 --> 00:09:08.690
So let's say this
is a million.

00:09:08.690 --> 00:09:10.520
So on day 0, we're
at 2 million.

00:09:10.520 --> 00:09:14.780
On day 1, we're at
1/4 of that.

00:09:14.780 --> 00:09:19.300
So let's say that's 1 million,
so then that is 500,000.

00:09:19.300 --> 00:09:23.190
On day 2, we're 1/4 of
this, so this is

00:09:23.190 --> 00:09:26.160
250,000, that's 125,000.

00:09:26.160 --> 00:09:28.770
And then on day 3, we're
1/4 of that.

00:09:28.770 --> 00:09:32.420
So you see the population of our
bacteria really goes down

00:09:32.420 --> 00:09:33.150
very quickly.

00:09:33.150 --> 00:09:35.250
It never gets quite to 0.

00:09:35.250 --> 00:09:37.310
We're always going to
have some bacteria.

00:09:37.310 --> 00:09:39.340
Well, eventually you're going
to have 1 bacteria, and you

00:09:39.340 --> 00:09:40.310
can get the antibiotic.

00:09:40.310 --> 00:09:42.690
And it's going to have a 1/4
chance of hitting it so,

00:09:42.690 --> 00:09:46.690
maybe, eventually you'll get to
0, but it slowly gets to 0.

00:09:46.690 --> 00:09:51.900
But the absolute number
decreases very, very fast.

00:09:51.900 --> 00:09:55.850
Now they also ask us, find
the size of the bacteria

00:09:55.850 --> 00:09:59.660
population 10 days after the
drug was first taken.

00:10:05.250 --> 00:10:06.960
So that's just n
is equal to 10.

00:10:06.960 --> 00:10:09.870
So it's going to be equal
to 1/4 to the tenth

00:10:09.870 --> 00:10:13.300
power times 2 million.

00:10:13.300 --> 00:10:17.000
Let's take our calculator out
to calculate to this.

00:10:17.000 --> 00:10:25.490
So let's take 1 divided by
4 to the tenth power.

00:10:25.490 --> 00:10:30.080
That's equal to this very small
number times 2 million.

00:10:30.080 --> 00:10:35.200
1, 2, 3, 1, 2, 3, is equal to--
we're only going to have

00:10:35.200 --> 00:10:40.310
a 1.9 bacteria after
the tenth day.

00:10:40.310 --> 00:10:45.720
This is going to be equal to 1.9
bacteria, so pretty much

00:10:45.720 --> 00:10:51.120
most of the bacteria is
gone by the tenth day.


Exponential growth functions

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:03.640
In this video, I want to
introduce you to the idea of

00:00:03.640 --> 00:00:11.070
an exponential function and
really just show you how fast

00:00:11.070 --> 00:00:12.660
these things can grow.

00:00:12.660 --> 00:00:16.340
So let's just write an example
exponential function here.

00:00:16.340 --> 00:00:22.500
So let's say we have y is equal
to 3 to the x power.

00:00:22.500 --> 00:00:24.575
Notice, this isn't x to
the third power, this

00:00:24.575 --> 00:00:26.260
is 3 to the x power.

00:00:26.260 --> 00:00:30.240
Our independent variable x
is the actual exponent.

00:00:30.240 --> 00:00:33.660
So let's make a table here to
see how quickly this thing

00:00:33.660 --> 00:00:37.740
grows, and maybe we'll
graph it as well.

00:00:37.740 --> 00:00:40.540
So let's take some
x values here.

00:00:40.540 --> 00:00:42.590
Let's start with x is
equal to negative 4.

00:00:42.590 --> 00:00:50.350
Then we'll go to negative 3,
negative 2, 0, 1, 2, 3, and 4.

00:00:50.350 --> 00:00:53.370
And let's figure out what our
y-values are going to be for

00:00:53.370 --> 00:00:55.540
each of these x-values.

00:00:55.540 --> 00:00:59.110
Now, here, y is going to be 3 to
the negative 4 power, which

00:00:59.110 --> 00:01:02.460
is equal to 1 over 3 to
the fourth power.

00:01:02.460 --> 00:01:06.420
3 to the third is 27 times
3 again is 81.

00:01:06.420 --> 00:01:10.670
So this is equal to 1/81.

00:01:10.670 --> 00:01:13.620
When x is equal to negative
3, y is 3.

00:01:13.620 --> 00:01:14.850
We'll do this in a
different color.

00:01:14.850 --> 00:01:16.250
This color is hard to read.

00:01:16.250 --> 00:01:18.530
y is 3 to the negative
3 power.

00:01:18.530 --> 00:01:21.840
Well, that's 1 over 3
to the third power,

00:01:21.840 --> 00:01:24.500
which is equal to 1/27.

00:01:24.500 --> 00:01:27.050
So we're going from a
super-small number to a less

00:01:27.050 --> 00:01:28.300
super-small number.

00:01:28.300 --> 00:01:31.460
And then 3 to the negative
2 power is

00:01:31.460 --> 00:01:34.470
going to be 1/9, right?

00:01:34.470 --> 00:01:39.970
1 over 3 squared, and then we
have 3 to the 0 power, which

00:01:39.970 --> 00:01:41.880
is just equal to 1.

00:01:41.880 --> 00:01:44.110
So we're getting a little bit
larger, a little bit larger,

00:01:44.110 --> 00:01:46.580
but you'll see that we
are about to explode.

00:01:46.580 --> 00:01:48.750
Now, we have 3 to
the first power.

00:01:48.750 --> 00:01:50.370
That's equal to 3.

00:01:50.370 --> 00:01:52.825
So we have 3 to the second
power, right? y is equal to 3

00:01:52.825 --> 00:01:53.830
to the second power.

00:01:53.830 --> 00:01:55.200
That's 9.

00:01:55.200 --> 00:01:57.050
3 to the third power, 27.

00:01:57.050 --> 00:01:58.820
3 to the fourth power, 81.

00:01:58.820 --> 00:02:02.970
If we were to put the
fifth power, 243.

00:02:02.970 --> 00:02:05.690
Let's graph this, just
to get an idea of how

00:02:05.690 --> 00:02:07.950
quickly we're exploding.

00:02:07.950 --> 00:02:09.495
Let me draw my axes here.

00:02:13.780 --> 00:02:17.270
So that's my x-axis and
that is my y-axis.

00:02:21.000 --> 00:02:24.400
And let me just do it in
increments of 5, because I

00:02:24.400 --> 00:02:28.500
really want to get the general
shape of the graph here.

00:02:28.500 --> 00:02:32.690
So let me just draw as straight
a line as I can.

00:02:32.690 --> 00:02:36.600
Let's say this is 5, 10, 15.

00:02:36.600 --> 00:02:38.240
Actually, I won't get
to 81 that way.

00:02:38.240 --> 00:02:41.250
I want to get to 81.

00:02:41.250 --> 00:02:42.480
Well, that's good enough.

00:02:42.480 --> 00:02:44.390
Let me draw it a little
bit differently

00:02:44.390 --> 00:02:46.030
than I've drawn it.

00:02:46.030 --> 00:02:49.960
So let me draw it down here
because all of these values,

00:02:49.960 --> 00:02:52.450
you might notice, are positive
values because I have a

00:02:52.450 --> 00:02:54.130
positive base.

00:02:54.130 --> 00:02:56.105
So let me draw it like this.

00:02:59.340 --> 00:03:00.330
Good enough.

00:03:00.330 --> 00:03:06.750
And then let's say I have 10,
20, 30, 40, 50, 60, 70, 80.

00:03:06.750 --> 00:03:08.320
That is 80 right there.

00:03:08.320 --> 00:03:09.500
That's 10.

00:03:09.500 --> 00:03:10.790
That's 30.

00:03:10.790 --> 00:03:12.860
That'll be good for
approximation.

00:03:12.860 --> 00:03:17.150
And then let's say that
this is negative 5.

00:03:17.150 --> 00:03:20.700
This is positive 5 right here.

00:03:20.700 --> 00:03:22.842
And actually, let me stretch
it out a little bit more.

00:03:22.842 --> 00:03:28.990
Let's say this is negative 1,
negative 2, negative 3,

00:03:28.990 --> 00:03:30.140
negative 4.

00:03:30.140 --> 00:03:34.830
Then we have 1, 2, 3, and 4.

00:03:34.830 --> 00:03:38.980
So when x is equal to 0, we're
equal to 1, right?

00:03:38.980 --> 00:03:42.840
When x is equal to 0, y is equal
to 1, which is right

00:03:42.840 --> 00:03:43.860
around there.

00:03:43.860 --> 00:03:47.460
When x is equal to 1, y is equal
to 3, which is right

00:03:47.460 --> 00:03:48.700
around there.

00:03:48.700 --> 00:03:51.770
When x is equal to 2, y is equal
to 9, which is right

00:03:51.770 --> 00:03:52.960
around there.

00:03:52.960 --> 00:03:56.440
When x equal to 3, y is
equal to 27, which is

00:03:56.440 --> 00:03:59.500
right around there.

00:03:59.500 --> 00:04:04.790
When x is equal to 4,
y is equal to 81.

00:04:04.790 --> 00:04:06.820
You see very quickly this
is just exploding.

00:04:06.820 --> 00:04:09.170
If I did 5, we'd go to
243, which wouldn't

00:04:09.170 --> 00:04:10.500
even fit on my screen.

00:04:10.500 --> 00:04:13.070
When you go to negative 1, we
get smaller and smaller.

00:04:13.070 --> 00:04:14.910
So at negative 1,
we're at 1/9.

00:04:14.910 --> 00:04:16.430
Eventually, you're not even
going to see this.

00:04:16.430 --> 00:04:18.760
It's going to get closer
and closer to zero.

00:04:18.760 --> 00:04:22.660
As this approaches larger and
larger negative numbers, or I

00:04:22.660 --> 00:04:25.220
guess I should say smaller
negative numbers, so 3 to the

00:04:25.220 --> 00:04:28.140
negative thousand, 3 to the
negative million, we're

00:04:28.140 --> 00:04:30.590
getting numbers closer and
closer to zero without

00:04:30.590 --> 00:04:32.490
actually ever approaching
zero.

00:04:32.490 --> 00:04:35.380
So as we go from negative
infinity, x is equal to

00:04:35.380 --> 00:04:38.060
negative infinity, we're getting
very close to zero,

00:04:38.060 --> 00:04:40.350
we're slowly getting our
way ourselves away from

00:04:40.350 --> 00:04:41.810
zero, but then bam!

00:04:41.810 --> 00:04:43.050
Once we start getting
into the positive

00:04:43.050 --> 00:04:46.440
numbers, we just explode.

00:04:46.440 --> 00:04:49.040
We just explode, and we
keep exploding at an

00:04:49.040 --> 00:04:50.660
ever-increasing rate.

00:04:50.660 --> 00:04:54.230
So the idea here is just to
show you that exponential

00:04:54.230 --> 00:04:58.850
functions are really,
really dramatic.

00:04:58.850 --> 00:05:02.240
Well, you can always construct
a faster expanding function.

00:05:02.240 --> 00:05:04.880
For example, you could say y is
equal to x to the x, even

00:05:04.880 --> 00:05:07.690
faster expanding, but out of the
ones that we deal with in

00:05:07.690 --> 00:05:10.970
everyday life, this is one of
the fastest. So given that,

00:05:10.970 --> 00:05:13.220
let's do some word problems
that just give us an

00:05:13.220 --> 00:05:17.250
appreciation for exponential
functions.

00:05:17.250 --> 00:05:18.970
So let's say that someone
sends out a chain

00:05:18.970 --> 00:05:22.162
letter in week 1.

00:05:22.162 --> 00:05:29.170
In week 1, someone sent a chain
letter to 10 people.

00:05:29.170 --> 00:05:33.130
And the chain letter says you
have to now send this chain

00:05:33.130 --> 00:05:36.775
letter to 10 more new people,
and if you don't, you're going

00:05:36.775 --> 00:05:40.140
to have bad luck, and your hair
is going to fall out, and

00:05:40.140 --> 00:05:42.350
you'll marry a frog,
or whatever else.

00:05:42.350 --> 00:05:45.730
So all of these people agree and
they go and each send it

00:05:45.730 --> 00:05:47.410
to 10 people the next week.

00:05:47.410 --> 00:05:51.610
So in week 2, they go
out and each send

00:05:51.610 --> 00:05:53.250
it to 10 more people.

00:05:53.250 --> 00:05:58.330
So each of those original 10
people are each sending out 10

00:05:58.330 --> 00:05:59.710
more of the letters.

00:05:59.710 --> 00:06:01.860
So now 100 people have
the letters, right?

00:06:01.860 --> 00:06:04.500
Each of those 10 sent
out 10, so that 100

00:06:04.500 --> 00:06:05.800
letters were sent out.

00:06:08.710 --> 00:06:10.050
10 were sent.

00:06:10.050 --> 00:06:12.510
Here, 100 were sent.

00:06:12.510 --> 00:06:14.570
In week 3, what's
going to happen?

00:06:14.570 --> 00:06:18.260
Each of those 100 people who
got this one, they're each

00:06:18.260 --> 00:06:20.730
going to send out 10, assuming
that everyone is really into

00:06:20.730 --> 00:06:22.110
chain letters.

00:06:22.110 --> 00:06:25.330
So 1,000 people are
going to get it.

00:06:25.330 --> 00:06:28.840
And so the general pattern
here is, the people who

00:06:28.840 --> 00:06:35.250
receive it, so in week n where
n is the week we're talking

00:06:35.250 --> 00:06:37.920
about, how many people
received the letter?

00:06:37.920 --> 00:06:46.960
In week n, we have 10 to the nth
people receive-- i before

00:06:46.960 --> 00:06:54.080
e except after c-- the letter.

00:06:54.080 --> 00:06:57.920
So if I were to ask you, how
many people are getting the

00:06:57.920 --> 00:06:59.700
letter on the sixth week?

00:07:03.370 --> 00:07:05.230
How many people are actually
going to receive that letter?

00:07:05.230 --> 00:07:07.140
Well, what's 10 to
the sixth power?

00:07:07.140 --> 00:07:12.150
10 to the sixth is equal to 1
with six zeroes, which is 1

00:07:12.150 --> 00:07:15.870
million people are going to
receive that letter in just 6

00:07:15.870 --> 00:07:18.250
weeks, just sending out
10 letters each.

00:07:18.250 --> 00:07:20.420
And obviously, in the real
world, most people chuck these

00:07:20.420 --> 00:07:23.170
in the basket, so you don't have
this good of a hit rate.

00:07:23.170 --> 00:07:26.290
But if you did, if every 10
people you sent it to also

00:07:26.290 --> 00:07:28.760
sent it to 10 people and so on
and so forth, by the sixth

00:07:28.760 --> 00:07:31.420
week, you would have
a million people.

00:07:31.420 --> 00:07:35.000
And by the ninth week, you would
have a billion people.

00:07:35.000 --> 00:07:36.610
And frankly, the week
after that, you

00:07:36.610 --> 00:07:38.100
would run out of people.

00:07:38.100 --> 00:07:40.250
I'll see you in the
next video.


Scientific notation examples

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.500
There are two whole Khan
Academy videos on what

00:00:03.500 --> 00:00:07.320
scientific notation is, why
we even worry about it.

00:00:07.320 --> 00:00:08.800
And it also goes through
a few examples.

00:00:08.800 --> 00:00:12.770
And what I want to do in this
video is just use a ck12.org

00:00:12.770 --> 00:00:19.960
Algebra I book to do some more
scientific notation examples.

00:00:19.960 --> 00:00:22.460
So let's take some things
that are written

00:00:22.460 --> 00:00:23.690
in scientific notation.

00:00:23.690 --> 00:00:25.980
Just as a reminder, scientific
notation is useful because it

00:00:25.980 --> 00:00:29.180
allows us to write really
large, or really small

00:00:29.180 --> 00:00:33.220
numbers, in ways that are easy
for our brains to, one, write

00:00:33.220 --> 00:00:35.120
down, and two, understand.

00:00:35.120 --> 00:00:36.800
So let's write down
some numbers.

00:00:36.800 --> 00:00:43.900
So let's say I have 3.102
times 10 to the second.

00:00:43.900 --> 00:00:47.510
And I want to write it as
just a numerical value.

00:00:47.510 --> 00:00:49.720
It's in scientific
notation already.

00:00:49.720 --> 00:00:53.840
It's written as a product
with a power of 10.

00:00:53.840 --> 00:00:55.310
So how do I write this?

00:00:55.310 --> 00:00:56.750
It's just a numeral.

00:00:56.750 --> 00:00:58.520
Well, there's a slow way
and the fast way.

00:00:58.520 --> 00:01:00.410
The slow way is to say, well,
this is the same thing as

00:01:00.410 --> 00:01:09.540
3.102 times 100, which means if
you multiplied 3.102 times

00:01:09.540 --> 00:01:16.040
100, it'll be 3, 1, 0, 2,
with two 0's behind it.

00:01:16.040 --> 00:01:18.670
And then we have 1, 2, 3 numbers
behind the decimal

00:01:18.670 --> 00:01:25.260
point, and that'd be
the right answer.

00:01:25.260 --> 00:01:29.230
This is equal to 310.2.

00:01:29.230 --> 00:01:33.570
Now, a faster way to do this
is just to say, well, look,

00:01:33.570 --> 00:01:36.510
right now I have only the 3 in
front of the decimal point.

00:01:36.510 --> 00:01:40.180
When I take something times 10
to the second power, I'm

00:01:40.180 --> 00:01:44.070
essentially shifting the decimal
point 2 to the right.

00:01:44.070 --> 00:01:49.400
So 3.102 times 10 to the second
power is the same thing

00:01:49.400 --> 00:01:54.260
as-- if I shift the decimal
point 1, and then 2, because

00:01:54.260 --> 00:01:55.920
this is 10 to the second
power-- it's

00:01:55.920 --> 00:02:00.090
same thing as 310.2.

00:02:00.090 --> 00:02:01.920
So this might be a faster
way of doing it.

00:02:01.920 --> 00:02:05.490
Every time you multiply it by
10, you shift the decimal to

00:02:05.490 --> 00:02:06.840
the right by 1.

00:02:06.840 --> 00:02:08.919
Let's do another example.

00:02:08.919 --> 00:02:14.910
Let's say I had 7.4 times
10 to the fourth.

00:02:14.910 --> 00:02:17.030
Well, let's just do
this the fast way.

00:02:17.030 --> 00:02:19.560
Let's shift the decimal
4 to the right.

00:02:19.560 --> 00:02:23.560
So 7.4 times 10 to the fourth.

00:02:23.560 --> 00:02:26.400
Times 10 to the 1, you're
going to get 74.

00:02:26.400 --> 00:02:31.050
Then times 10 to the second,
you're going to get 740.

00:02:31.050 --> 00:02:33.080
We're going to have to add a
0 there, because we have to

00:02:33.080 --> 00:02:34.500
shift the decimal again.

00:02:34.500 --> 00:02:38.710
10 to the third, you're
going to have 7,400.

00:02:38.710 --> 00:02:42.390
And then 10 to the fourth,
you're going to have 74,000.

00:02:42.390 --> 00:02:43.825
Notice, I just took
this decimal and

00:02:43.825 --> 00:02:49.230
went 1, 2, 3, 4 spaces.

00:02:49.230 --> 00:02:53.130
So this is equal to 74,000.

00:02:53.130 --> 00:02:56.410
And when I had 74, and I had to
shift the decimal 1 more to

00:02:56.410 --> 00:02:57.920
the right, I had to
throw in a 0 here.

00:02:57.920 --> 00:03:00.820
I'm multiplying it by 10.

00:03:00.820 --> 00:03:06.760
Another way to think about it
is, I need 10 spaces between

00:03:06.760 --> 00:03:09.420
the leading digit
and the decimal.

00:03:09.420 --> 00:03:11.250
So right here, I only
have 1 space.

00:03:11.250 --> 00:03:15.820
I'll need 4 spaces,
So, 1, 2, 3, 4.

00:03:15.820 --> 00:03:17.670
Let's do a few more examples,
because I think the more

00:03:17.670 --> 00:03:20.230
examples, the more you'll
get what's going on.

00:03:20.230 --> 00:03:26.590
So I have 1.75 times 10
to the negative 3.

00:03:26.590 --> 00:03:29.050
This is in scientific notation,
and I want to just

00:03:29.050 --> 00:03:31.800
write the numerical
value of this.

00:03:31.800 --> 00:03:34.360
So when you take something to
the negative times 10 to the

00:03:34.360 --> 00:03:38.250
negative power, you shift
the decimal to the left.

00:03:38.250 --> 00:03:42.130
So this is 1.75.

00:03:42.130 --> 00:03:46.550
So if you do it times 10 to the
negative 1 power, you'll

00:03:46.550 --> 00:03:48.330
go 1 to the left.

00:03:48.330 --> 00:03:51.940
But if you do times 10 to the
negative 2 power, you'll go 2

00:03:51.940 --> 00:03:52.570
to the left.

00:03:52.570 --> 00:03:54.130
And you'd have to
put a 0 here.

00:03:54.130 --> 00:03:57.670
And if you do times 10 to the
negative 3, you'd go 3 to the

00:03:57.670 --> 00:04:01.160
left, and you would have
to add another 0.

00:04:01.160 --> 00:04:04.860
So you take this decimal and
go 1, 2, 3 to the left.

00:04:04.860 --> 00:04:12.440
So our answer would be 0.00175
is the same thing as 1.75

00:04:12.440 --> 00:04:14.560
times 10 to the negative 3.

00:04:14.560 --> 00:04:19.519
And another way to check that
you got the right answer is if

00:04:19.519 --> 00:04:23.870
you have a 1 right here, if you
count the 1, 1 including

00:04:23.870 --> 00:04:27.310
the 0's to the right of the
decimal should be the same as

00:04:27.310 --> 00:04:28.880
the negative exponent here.

00:04:28.880 --> 00:04:33.150
So you have 1, 2, 3 numbers
behind the decimal.

00:04:33.150 --> 00:04:35.520
That's the same thing as to
the negative 3 power.

00:04:35.520 --> 00:04:40.030
You're doing 1,000th, so this
is 1,000th right there.

00:04:40.030 --> 00:04:41.330
Let's do another example.

00:04:41.330 --> 00:04:42.230
Actually let's mix it up.

00:04:42.230 --> 00:04:45.710
Let's start with something
that's written as a numeral

00:04:45.710 --> 00:04:48.250
and then write it in scientific
notation.

00:04:48.250 --> 00:04:50.860
So let's say I have 120,000.

00:04:50.860 --> 00:04:53.430
So that's just its numerical
value, and I want to write it

00:04:53.430 --> 00:04:55.270
in scientific notation.

00:04:55.270 --> 00:05:01.490
So this I can write as-- I take
the leading digit-- 1.2

00:05:01.490 --> 00:05:05.910
times 10 to the-- and I just
count how many digits there

00:05:05.910 --> 00:05:07.680
are behind the leading digit.

00:05:07.680 --> 00:05:10.760
1, 2, 3, 4, 5.

00:05:10.760 --> 00:05:13.270
So 1.2 times 10 to the fifth.

00:05:13.270 --> 00:05:16.650
And if you want to internalize
why that makes sense, 10 to

00:05:16.650 --> 00:05:18.950
the fifth is 10,000.

00:05:18.950 --> 00:05:24.390
So 1.2-- 10 to the
fifth is 100,000.

00:05:24.390 --> 00:05:28.970
So it's 1.2 times--
1, 1, 2, 3, 4, 5.

00:05:28.970 --> 00:05:29.900
You have five 0's.

00:05:29.900 --> 00:05:31.580
That's 10 to the fifth.

00:05:31.580 --> 00:05:35.230
So 1.2 times 100,000 is
going to be a 120,000.

00:05:35.230 --> 00:05:38.890
It's going to be 1 and 1/5
times 100,000, so 120.

00:05:38.890 --> 00:05:41.400
So hopefully that's
sinking in.

00:05:41.400 --> 00:05:43.030
So let's do another one.

00:05:43.030 --> 00:05:49.840
Let's say the numerical
value is 1,765,244.

00:05:49.840 --> 00:05:54.150
I want to write this in
scientific notation, so I take

00:05:54.150 --> 00:05:57.430
the leading digit, 1,
put a decimal sign.

00:05:57.430 --> 00:05:59.050
Everything else goes
behind the decimal.

00:05:59.050 --> 00:06:02.530
7, 6, 5, 2, 4, 4.

00:06:02.530 --> 00:06:05.460
And then you count how many
digits there were between the

00:06:05.460 --> 00:06:07.920
leading digit, and I guess, you
could imagine, the first

00:06:07.920 --> 00:06:08.640
decimal sign.

00:06:08.640 --> 00:06:11.720
Because you could have numbers
that keep going over here.

00:06:11.720 --> 00:06:14.370
So between the leading digit
and the decimal sign.

00:06:14.370 --> 00:06:19.080
And you have 1, 2, 3,
4, 5, 6 digits.

00:06:19.080 --> 00:06:22.310
So this is times 10
to the sixth.

00:06:22.310 --> 00:06:23.620
And 10 to the sixth
is just 1 million.

00:06:23.620 --> 00:06:28.200
So it's 1.765244 times 1
million, which makes sense.

00:06:28.200 --> 00:06:32.750
Roughly 1.7 times million
is roughly 1.7 million.

00:06:32.750 --> 00:06:34.260
This is a little bit
more than 1.7

00:06:34.260 --> 00:06:36.970
million, so it makes sense.

00:06:36.970 --> 00:06:38.360
Let's do another one.

00:06:38.360 --> 00:06:40.730
How do I write 12 in scientific
notation?

00:06:40.730 --> 00:06:42.120
Same drill.

00:06:42.120 --> 00:06:46.430
It's equal to 1.2 times-- well,
we only have 1 digit

00:06:46.430 --> 00:06:49.875
between the 1 and the decimal
spot, or the decimal point.

00:06:49.875 --> 00:06:54.900
So it's 1.2 times 10 to the
first power, or 1.2 times 10,

00:06:54.900 --> 00:06:57.540
which is definitely
equal to 12.

00:06:57.540 --> 00:07:00.230
Let's do a couple of examples
where we're taking 10 to a

00:07:00.230 --> 00:07:01.530
negative power.

00:07:01.530 --> 00:07:08.480
So let's say we had 0.00281, and
we want to write this in

00:07:08.480 --> 00:07:10.450
scientific notation.

00:07:10.450 --> 00:07:15.850
So what you do, is you just have
to think, well, how many

00:07:15.850 --> 00:07:21.010
digits are there to include
the leading

00:07:21.010 --> 00:07:22.365
numeral in the value?

00:07:22.365 --> 00:07:26.750
So what I mean there
is count, 1, 2, 3.

00:07:26.750 --> 00:07:27.980
So what we want to
do is we move the

00:07:27.980 --> 00:07:32.040
decimal 1, 2, 3 spaces.

00:07:32.040 --> 00:07:34.710
So one way you could think about
it is, you can multiply.

00:07:34.710 --> 00:07:37.220
To move the decimal to the
right 3 spaces, you would

00:07:37.220 --> 00:07:40.200
multiply it by 10
to the third.

00:07:40.200 --> 00:07:42.170
But if you're multiplying
something by 10 to the third,

00:07:42.170 --> 00:07:43.340
you're changing its values.

00:07:43.340 --> 00:07:47.440
So we also have to multiply
by 10 to the negative 3.

00:07:47.440 --> 00:07:50.090
Only this way will you not
change the value, right?

00:07:50.090 --> 00:07:53.920
If I multiply by 10 to the 3,
times 10 to the negative 2-- 3

00:07:53.920 --> 00:07:57.440
minus 3 is 0-- this is just
like multiplying it by 1.

00:07:57.440 --> 00:07:59.820
So what is this going
to equal?

00:07:59.820 --> 00:08:03.420
If I take the decimal and I move
it 3 spaces to the right,

00:08:03.420 --> 00:08:07.510
this part right here is going
to be equal to 2.81.

00:08:07.510 --> 00:08:10.880
And then we're left with
this one, times 10 to

00:08:10.880 --> 00:08:12.720
the negative 3.

00:08:12.720 --> 00:08:15.770
Now, a very quick way to do it
is just to say, look, let me

00:08:15.770 --> 00:08:18.860
count-- including the leading
numeral-- how many spaces I

00:08:18.860 --> 00:08:19.970
have behind the decimal.

00:08:19.970 --> 00:08:21.440
1, 2, 3.

00:08:21.440 --> 00:08:25.060
So it's going to be 2.81
times 10 to the

00:08:25.060 --> 00:08:28.430
negative 1, 2, 3 power.

00:08:28.430 --> 00:08:30.470
Let's do one more like that.

00:08:30.470 --> 00:08:33.080
Let me actually scroll
up here.

00:08:33.080 --> 00:08:34.520
Let's do one more like that.

00:08:34.520 --> 00:08:43.240
Let's say I have 1, 2, 3, 4, 5,
6-- how many 0's do I have

00:08:43.240 --> 00:08:43.750
in this problem?

00:08:43.750 --> 00:08:45.030
Well, I'll just make
up something.

00:08:45.030 --> 00:08:47.240
0, 2, 7.

00:08:47.240 --> 00:08:49.830
And you wanted to write that
in scientific notation.

00:08:49.830 --> 00:08:51.540
Well, you count all the
digits up to the

00:08:51.540 --> 00:08:53.380
2, behind the decimal.

00:08:53.380 --> 00:08:57.610
So 1, 2, 3, 4, 5, 6, 7, 8.

00:08:57.610 --> 00:09:01.870
So this is going to
be 2.7 times 10 to

00:09:01.870 --> 00:09:04.460
the negative 8 power.

00:09:04.460 --> 00:09:08.270
Now let's do another one,
where we start with the

00:09:08.270 --> 00:09:10.840
scientific notation value
and we want to go

00:09:10.840 --> 00:09:12.760
to the numeric value.

00:09:12.760 --> 00:09:13.690
Just to mix things up.

00:09:13.690 --> 00:09:17.190
So let's say you have
2.9 times 10 to

00:09:17.190 --> 00:09:18.830
the negative fifth.

00:09:18.830 --> 00:09:22.360
So one way to think about is,
this leading numeral, plus all

00:09:22.360 --> 00:09:25.110
0's to the left of the
decimal spot, is

00:09:25.110 --> 00:09:26.980
going to be five digits.

00:09:26.980 --> 00:09:29.600
So you have a 2 and a 9,
and then you're going

00:09:29.600 --> 00:09:33.170
to have 4 more 0's.

00:09:33.170 --> 00:09:35.450
1, 2, 3, 4.

00:09:35.450 --> 00:09:37.140
And then you're going to
have your decimal.

00:09:37.140 --> 00:09:39.610
And how did I know 4 0's?

00:09:39.610 --> 00:09:45.370
Because I'm counting,, this is
1, 2, 3, 4, 5 spaces behind

00:09:45.370 --> 00:09:48.510
the decimal, including
the leading numeral.

00:09:48.510 --> 00:09:52.600
And so it's 0.000029.

00:09:52.600 --> 00:09:54.670
And just to verify, do
the other technique.

00:09:54.670 --> 00:09:57.110
How do I write this in
scientific notation?

00:09:57.110 --> 00:10:03.310
I count all of the digits, all
of the leading 0's behind the

00:10:03.310 --> 00:10:07.050
decimal, including the leading
non-zero numeral.

00:10:07.050 --> 00:10:10.330
So I have 1, 2, 3,
4, 5 digits.

00:10:10.330 --> 00:10:12.725
So it's 10 to the negative 5.

00:10:12.725 --> 00:10:20.590
And so it'll be 2.9 times
10 to the negative 5.

00:10:20.590 --> 00:10:23.230
And once again, this isn't
just some type

00:10:23.230 --> 00:10:24.450
of black magic here.

00:10:24.450 --> 00:10:27.270
This actually makes
a lot of sense.

00:10:27.270 --> 00:10:30.920
If I wanted to get this number
to 2.9, what I would have to

00:10:30.920 --> 00:10:37.310
do is move the decimal over 1,
2, 3, 4, 5 spots, like that.

00:10:37.310 --> 00:10:43.010
And to get the decimal to move
over the right by 5 spots--

00:10:43.010 --> 00:10:47.700
let's just say with
0, 0, 0, 0, 2, 9.

00:10:47.700 --> 00:10:51.670
If I multiply it by 10 to the
fifth, I'm also going to have

00:10:51.670 --> 00:10:54.700
to multiply it by 10
to the negative 5.

00:10:54.700 --> 00:10:56.230
So I don't want to change
the number.

00:10:56.230 --> 00:10:59.060
This right here is just
multiplying something by 1.

00:10:59.060 --> 00:11:02.790
10 to the fifth times 10
to the negative 5 is 1.

00:11:02.790 --> 00:11:07.910
So this right here is
essentially going to move the

00:11:07.910 --> 00:11:09.850
decimal 5 to the right.

00:11:09.850 --> 00:11:12.700
1, 2, 3, 4, 5.

00:11:12.700 --> 00:11:16.660
So this will be 2.5, and then
we're going to be left with

00:11:16.660 --> 00:11:20.310
times 10 to the negative 5.

00:11:20.310 --> 00:11:22.640
Anyway, hopefully, you
found that scientific

00:11:22.640 --> 00:11:24.980
notation drill useful.