Polynomials intro

WEBVTT
Kind: captions
Language: en

00:00:00.773 --> 00:00:05.687
- [Sal] Let's explore the
notion of a polynomial.

00:00:05.687 --> 00:00:07.677
This seems like a very complicated word,

00:00:07.677 --> 00:00:09.464
but if you break it down
it'll start to make sense,

00:00:09.464 --> 00:00:13.285
especially when we start to
see examples of polynomials.

00:00:13.285 --> 00:00:15.440
The first part of this word,

00:00:15.440 --> 00:00:17.791
lemme underline it, we have poly.

00:00:17.791 --> 00:00:20.621
This comes from Greek, for many.

00:00:20.621 --> 00:00:23.913
You see poly a lot in
the English language,

00:00:23.913 --> 00:00:26.432
referring to the notion
of many of something.

00:00:26.432 --> 00:00:29.282
In this case, it's many nomials.

00:00:29.282 --> 00:00:33.631
Nomial comes from Latin, from
the Latin nomen, for name.

00:00:33.631 --> 00:00:36.271
You could view this as many names.

00:00:36.271 --> 00:00:37.897
But in a mathematical context,

00:00:37.897 --> 00:00:40.298
it's really referring to many terms.

00:00:40.298 --> 00:00:41.882
We're gonna talk, in a little bit,

00:00:41.882 --> 00:00:44.240
about what a term really is.

00:00:44.240 --> 00:00:47.207
But to get a tangible sense
of what are polynomials

00:00:47.207 --> 00:00:50.378
and what are not polynomials,
lemme give you some examples.

00:00:50.378 --> 00:00:51.959
And then we could write some, maybe,

00:00:51.959 --> 00:00:54.313
more formal rules for them.

00:00:54.313 --> 00:00:57.771
So, an example of a polynomial could be

00:00:57.771 --> 00:01:01.771
10x to the seventh power
minus nine x squared

00:01:02.862 --> 00:01:05.529
plus 15x to the third plus nine.

00:01:07.419 --> 00:01:09.492
This is a polynomial.

00:01:09.492 --> 00:01:11.686
Another example of a polynomial.

00:01:11.686 --> 00:01:13.853
Nine a squared minus five.

00:01:16.721 --> 00:01:18.549
Even if I just have one number,

00:01:18.549 --> 00:01:21.843
even if I were to just
write the number six,

00:01:21.843 --> 00:01:24.973
that can officially be
considered a polynomial.

00:01:24.973 --> 00:01:29.140
If I were to write seven
x squared minus three.

00:01:30.183 --> 00:01:31.118
Lemme do it another variable.

00:01:31.118 --> 00:01:34.285
Seven y squared minus three y plus pi,

00:01:36.042 --> 00:01:38.810
that, too, would be a polynomial.

00:01:38.810 --> 00:01:40.356
These are examples of polynomials.

00:01:40.356 --> 00:01:43.890
What are examples of things
that are not polynomials?

00:01:43.890 --> 00:01:48.078
Well, if I were to
replace the seventh power

00:01:48.078 --> 00:01:50.151
right over here with a
negative seven power.

00:01:50.151 --> 00:01:53.964
If I were to write 10x to
the negative seven power

00:01:53.964 --> 00:01:58.131
minus nine x squared plus 15x
to the third power plus nine,

00:02:00.630 --> 00:02:02.418
this would not be a polynomial.

00:02:02.418 --> 00:02:04.249
So I think you might
be sensing a rule here

00:02:04.249 --> 00:02:06.133
for what makes something a polynomial.

00:02:06.133 --> 00:02:09.201
You have to have nonnegative powers

00:02:09.201 --> 00:02:12.250
of your variable in each of the terms.

00:02:12.250 --> 00:02:14.524
I just used that word,
terms, so lemme explain it,

00:02:14.524 --> 00:02:17.246
'cause it'll help me explain
what a polynomial is.

00:02:17.246 --> 00:02:22.079
A polynomial is something that
is made up of a sum of terms.

00:02:22.079 --> 00:02:25.332
And so, for example, in
this first polynomial,

00:02:25.332 --> 00:02:27.931
the first term is 10x to the seventh;

00:02:27.931 --> 00:02:31.344
the second term is
negative nine x squared;

00:02:31.344 --> 00:02:33.781
the next term is 15x to the third;

00:02:33.781 --> 00:02:34.880
and then the last term,

00:02:34.880 --> 00:02:37.561
maybe you could say the
fourth term, is nine.

00:02:37.561 --> 00:02:38.691
You can see something.

00:02:38.691 --> 00:02:40.918
Let me underline these.

00:02:40.918 --> 00:02:42.623
These are all terms.

00:02:42.623 --> 00:02:46.811
This is a four-term
polynomial right over here.

00:02:46.811 --> 00:02:48.637
You could say: "Hey, wait,
this thing you wrote in red,

00:02:48.637 --> 00:02:50.304
"this also has four terms."

00:02:50.304 --> 00:02:52.130
We have to put a few more rules for it

00:02:52.130 --> 00:02:53.350
to officially be a polynomial,

00:02:53.350 --> 00:02:55.873
especially a polynomial in one variable.

00:02:55.873 --> 00:02:57.296
Each of those terms are going

00:02:57.296 --> 00:03:00.102
to be made up of a coefficient.

00:03:00.102 --> 00:03:01.973
This is the thing that multiplies

00:03:01.973 --> 00:03:04.492
the variable to some power.

00:03:04.492 --> 00:03:07.641
So in this first term
the coefficient is 10.

00:03:07.641 --> 00:03:10.606
Lemme write this word down, coefficient.

00:03:10.606 --> 00:03:13.000
It's another fancy word,
but it's just a thing

00:03:13.000 --> 00:03:14.706
that's multiplied, in this case,

00:03:14.706 --> 00:03:17.518
times the variable, which
is x to seventh power.

00:03:17.518 --> 00:03:19.488
The first coefficient is 10.

00:03:19.488 --> 00:03:20.708
The next coefficient.

00:03:20.708 --> 00:03:22.543
Actually, lemme be careful here,

00:03:22.543 --> 00:03:26.710
because the second coefficient
here is negative nine.

00:03:27.653 --> 00:03:30.236
We are looking at coefficients.

00:03:31.271 --> 00:03:33.873
The third coefficient here is 15.

00:03:33.873 --> 00:03:37.207
You can view this fourth
term, or this fourth number,

00:03:37.207 --> 00:03:40.903
as the coefficient because
this could be rewritten as,

00:03:40.903 --> 00:03:42.004
instead of just writing as nine,

00:03:42.004 --> 00:03:45.885
you could write it as
nine x to the zero power.

00:03:45.885 --> 00:03:48.404
And then it looks a little bit
clearer, like a coefficient.

00:03:48.404 --> 00:03:51.901
So, in general, a polynomial is the sum

00:03:51.901 --> 00:03:55.484
of a finite number of
terms where each term

00:03:56.581 --> 00:03:59.226
has a coefficient, which I could represent

00:03:59.226 --> 00:04:03.297
with the letter A, being
multiplied by a variable

00:04:03.297 --> 00:04:06.964
being raised to a
nonnegative integer power.

00:04:09.999 --> 00:04:13.654
So, this right over here is a coefficient.

00:04:13.654 --> 00:04:15.812
It can be, if we're dealing...

00:04:15.812 --> 00:04:17.360
Well, I don't wanna get too technical.

00:04:17.360 --> 00:04:18.730
Positive, negative number.

00:04:18.730 --> 00:04:20.855
Could be any real number.

00:04:20.855 --> 00:04:22.239
We have our variable.

00:04:22.239 --> 00:04:26.406
And then the exponent,
here, has to be nonnegative.

00:04:28.014 --> 00:04:28.931
Nonnegative

00:04:30.516 --> 00:04:31.349
integer.

00:04:32.603 --> 00:04:34.475
So here, the reason
why what I wrote in red

00:04:34.475 --> 00:04:36.595
is not a polynomial is because here I have

00:04:36.595 --> 00:04:39.930
an exponent that is a negative integer.

00:04:39.930 --> 00:04:41.355
Let's give some other examples

00:04:41.355 --> 00:04:44.201
of things that are not polynomials.

00:04:44.201 --> 00:04:47.701
So, if I were to change the second one to,

00:04:49.041 --> 00:04:50.301
instead of nine a squared,

00:04:50.301 --> 00:04:54.815
if I wrote it as nine a to
the one half power minus five,

00:04:54.815 --> 00:04:57.335
this is not a polynomial
because this exponent

00:04:57.335 --> 00:05:00.668
right over here, it is
no longer an integer;

00:05:00.668 --> 00:05:02.250
it's one half.

00:05:02.250 --> 00:05:05.137
This is the same thing as nine times

00:05:05.137 --> 00:05:07.739
the square root of a minus five.

00:05:07.739 --> 00:05:10.739
This also would not be a polynomial.

00:05:11.885 --> 00:05:16.052
Or, if I were to write nine
a to the a power minus five,

00:05:18.553 --> 00:05:21.315
also not a polynomial
because here the exponent

00:05:21.315 --> 00:05:25.148
is a variable; it's not
a nonnegative integer.

00:05:26.642 --> 00:05:30.059
All of these are examples of polynomials.

00:05:31.036 --> 00:05:33.879
There's a few more pieces of terminology

00:05:33.879 --> 00:05:36.046
that are valuable to know.

00:05:38.036 --> 00:05:41.483
Polynomial is a general term
for one of these expression

00:05:41.483 --> 00:05:44.525
that has multiple terms, a finite number,

00:05:44.525 --> 00:05:46.108
so not an infinite number,

00:05:46.108 --> 00:05:48.751
and each of the terms has this form.

00:05:48.751 --> 00:05:50.334
But there's more specific terms for when

00:05:50.334 --> 00:05:54.929
you have only one term or
two terms or three terms.

00:05:54.929 --> 00:05:58.659
When you have one term,
it's called a monomial.

00:05:58.659 --> 00:06:00.242
This is a monomial.

00:06:02.592 --> 00:06:04.177
This is an example of a monomial,

00:06:04.177 --> 00:06:06.874
which we could write as six x to the zero.

00:06:06.874 --> 00:06:10.776
Another example of a monomial might be

00:06:10.776 --> 00:06:12.609
10z to the 15th power.

00:06:14.754 --> 00:06:16.339
That's also a monomial.

00:06:16.339 --> 00:06:18.700
Your coefficient could be pi.

00:06:18.700 --> 00:06:19.533
Pi.

00:06:19.533 --> 00:06:20.529
Whoops.

00:06:20.529 --> 00:06:21.812
Could be pi.

00:06:21.812 --> 00:06:25.812
So we could write pi times
b to the fifth power.

00:06:28.271 --> 00:06:30.921
Any of these would be monomials.

00:06:30.921 --> 00:06:32.424
So what's a binomial?

00:06:32.424 --> 00:06:33.849
Binomial's where you have two terms.

00:06:33.849 --> 00:06:36.129
Monomial, mono for one, one term.

00:06:36.129 --> 00:06:38.161
Binomial is you have two terms.

00:06:38.161 --> 00:06:41.167
This right over here is a binomial.

00:06:41.167 --> 00:06:42.345
Binomial.

00:06:42.345 --> 00:06:44.132
You have two terms.

00:06:44.132 --> 00:06:47.590
All these are polynomials but
these are subclassifications.

00:06:47.590 --> 00:06:50.339
It's a binomial; you have one, two terms.

00:06:50.339 --> 00:06:53.506
Another example of a binomial would be

00:06:55.069 --> 00:06:57.819
three y to the third plus five y.

00:06:58.797 --> 00:07:00.625
Once again, you have two terms

00:07:00.625 --> 00:07:03.358
that have this form right over here.

00:07:03.358 --> 00:07:05.889
You'll also hear the term trinomial.

00:07:05.889 --> 00:07:08.408
Trinomial's when you have three terms.

00:07:08.408 --> 00:07:09.241
Trinomial.

00:07:11.499 --> 00:07:12.999
This right over here is an example.

00:07:12.999 --> 00:07:14.340
This is the first term;

00:07:14.340 --> 00:07:16.006
this is the second term;

00:07:16.006 --> 00:07:18.256
and this is the third term.

00:07:19.223 --> 00:07:22.616
Now, the next word that
you will hear often

00:07:22.616 --> 00:07:26.192
in the context with
polynomials is the notion

00:07:26.192 --> 00:07:29.486
of the degree of a polynomial.

00:07:29.486 --> 00:07:30.993
You might hear people say:

00:07:30.993 --> 00:07:33.590
"What is the degree of a polynomial?",

00:07:33.590 --> 00:07:37.396
or "What is the degree of a
given term of a polynomial?"

00:07:37.396 --> 00:07:39.924
Let's start with the
degree of a given term.

00:07:39.924 --> 00:07:41.299
Let's go to this polynomial here.

00:07:41.299 --> 00:07:44.024
We have this first term,
10x to the seventh.

00:07:44.024 --> 00:07:47.390
The degree is the power that
we're raising the variable to.

00:07:47.390 --> 00:07:49.669
So this is a seventh-degree term.

00:07:49.669 --> 00:07:52.393
The second term is a second-degree term.

00:07:52.393 --> 00:07:55.648
The third term is a third-degree term.

00:07:55.648 --> 00:07:57.273
And you could view this constant term,

00:07:57.273 --> 00:07:59.752
which is really just nine,
you could view that as,

00:07:59.752 --> 00:08:01.583
sometimes people say the constant term.

00:08:01.583 --> 00:08:05.282
Sometimes people will
say the zero-degree term.

00:08:05.282 --> 00:08:07.010
If people are talking about the degree

00:08:07.010 --> 00:08:09.203
of the entire polynomial,
they're gonna say:

00:08:09.203 --> 00:08:11.842
"What is the degree of the highest term?

00:08:11.842 --> 00:08:14.164
"What is the term with
the highest degree?"

00:08:14.164 --> 00:08:17.418
That degree will be the degree
of the entire polynomial.

00:08:17.418 --> 00:08:19.244
So, this first polynomial,

00:08:19.244 --> 00:08:22.047
this is a seventh-degree polynomial.

00:08:22.047 --> 00:08:25.859
This one right over here is
a second-degree polynomial

00:08:25.859 --> 00:08:27.315
because it has a second-degree term

00:08:27.315 --> 00:08:29.753
and that's the highest-degree term.

00:08:29.753 --> 00:08:32.390
This right over here is a third-degree.

00:08:32.390 --> 00:08:34.952
You could even say third-degree binomial

00:08:34.952 --> 00:08:38.287
because its highest-degree
term has degree three.

00:08:38.287 --> 00:08:41.248
If this said five y to the
seventh instead of five y,

00:08:41.248 --> 00:08:44.095
then it would be a
seventh-degree binomial.

00:08:44.095 --> 00:08:47.916
This right over here is
a 15th-degree monomial.

00:08:47.916 --> 00:08:50.749
This is a second-degree trinomial.

00:08:56.519 --> 00:08:58.548
A few more things I will introduce you to

00:08:58.548 --> 00:09:02.121
is the idea of a leading term
and a leading coefficient.

00:09:02.121 --> 00:09:03.178
Lemme write this down.

00:09:03.178 --> 00:09:06.678
The notion of what it means to be leading.

00:09:10.173 --> 00:09:14.322
It can mean whatever is the
first term or the coefficient.

00:09:14.322 --> 00:09:16.836
If you're saying leading
term, it's the first term.

00:09:16.836 --> 00:09:18.140
If you're saying leading coefficient,

00:09:18.140 --> 00:09:20.191
it's the coefficient in the first term.

00:09:20.191 --> 00:09:22.428
But it's oftentimes associated with

00:09:22.428 --> 00:09:26.788
a polynomial being
written in standard form.

00:09:26.788 --> 00:09:27.955
Standard form.

00:09:29.933 --> 00:09:31.016
Standard form

00:09:32.942 --> 00:09:35.997
is where you write the
terms in degree order,

00:09:35.997 --> 00:09:39.003
starting with the highest-degree term.

00:09:39.003 --> 00:09:41.362
So, for example, what I have up here,

00:09:41.362 --> 00:09:43.562
this is not in standard form;

00:09:43.562 --> 00:09:45.960
because I do have the
highest-degree term first,

00:09:45.960 --> 00:09:47.382
but then I should go to the next highest,

00:09:47.382 --> 00:09:48.884
which is the x to the third.

00:09:48.884 --> 00:09:51.160
But here I wrote x squared
next, so this is not standard.

00:09:51.160 --> 00:09:53.195
If I wanted to write it in standard form,

00:09:53.195 --> 00:09:55.717
it would be 10x to the seventh power,

00:09:55.717 --> 00:09:58.887
which is the highest-degree
term, has degree seven.

00:09:58.887 --> 00:10:01.039
Then, 15x to the third.

00:10:01.039 --> 00:10:04.858
So, plus 15x to the third, which
is the next highest degree.

00:10:04.858 --> 00:10:08.108
Then, negative nine x squared

00:10:08.108 --> 00:10:09.897
is the next highest degree term.

00:10:09.897 --> 00:10:12.342
And then, the lowest-degree
term here is plus nine,

00:10:12.342 --> 00:10:14.456
or plus nine x to zero.

00:10:14.456 --> 00:10:16.367
Now this is in standard form.

00:10:16.367 --> 00:10:19.131
I have written the terms in
order of decreasing degree,

00:10:19.131 --> 00:10:21.045
with the highest degree first.

00:10:21.045 --> 00:10:25.112
Here, it's clear that your leading term

00:10:25.112 --> 00:10:29.265
is 10x to the seventh,
'cause it's the first one,

00:10:29.265 --> 00:10:33.432
and our leading coefficient
here is the number 10.

00:10:34.377 --> 00:10:36.170
So, there was a lot in that video,

00:10:36.170 --> 00:10:38.205
but hopefully the notion of a polynomial

00:10:38.205 --> 00:10:40.966
isn't seeming too
intimidating at this point.

00:10:40.966 --> 00:10:43.446
These are really useful
words to be familiar with

00:10:43.446 --> 00:10:46.779
as you continue on on your math journey.


Introduction to exponential decay

WEBVTT
Kind: captions
Language: en

00:00:00.584 --> 00:00:01.784
- [Narrator] What we're
going to do in this video

00:00:01.784 --> 00:00:04.616
is quickly review exponential growth

00:00:04.616 --> 00:00:07.161
and then use that as our platform

00:00:07.161 --> 00:00:11.120
to introduce ourselves
to exponential decay.

00:00:11.120 --> 00:00:13.416
So let's review exponential growth.

00:00:13.416 --> 00:00:15.989
Let's say we have something that,

00:00:15.989 --> 00:00:18.739
and I'll do this on a table here.

00:00:19.647 --> 00:00:22.016
Just gonna make that straight.

00:00:22.016 --> 00:00:24.988
So let's say this is
our x and this is our y.

00:00:24.988 --> 00:00:29.126
Now let's say when x is
zero, y is equal to three.

00:00:29.126 --> 00:00:32.313
And every time we increase
x by 1, we double y.

00:00:32.313 --> 00:00:35.230
So y is gonna go from three to six.

00:00:36.585 --> 00:00:39.117
If x increases by one
again, so we go to two,

00:00:39.117 --> 00:00:41.520
we're gonna double y again.

00:00:41.520 --> 00:00:43.770
And so six times two is 12.

00:00:44.707 --> 00:00:48.370
This right over here
is exponential growth.

00:00:48.370 --> 00:00:50.832
And you could even go for negative x's.

00:00:50.832 --> 00:00:54.129
When x is negative one, well,
if we're going back one in x,

00:00:54.129 --> 00:00:56.004
we would divide by two.

00:00:56.004 --> 00:00:58.424
So this is going to be 3/2.

00:00:58.424 --> 00:00:59.394
3/2.

00:00:59.394 --> 00:01:01.704
And notice if you go from
negative one to zero,

00:01:01.704 --> 00:01:04.793
you once again, you
keep multiplying by two

00:01:04.793 --> 00:01:06.731
and this will keep on happening.

00:01:06.731 --> 00:01:09.332
And you can describe
this with an equation.

00:01:09.332 --> 00:01:12.320
You could say that y is equal to,

00:01:12.320 --> 00:01:15.100
and sometimes people might
call this your y intercept

00:01:15.100 --> 00:01:16.623
or your initial value,

00:01:16.623 --> 00:01:19.340
is equal to three,
essentially what happens

00:01:19.340 --> 00:01:22.989
when x equals zero, is
equal to three times

00:01:22.989 --> 00:01:25.099
our common ratio, and our common ratio is,

00:01:25.099 --> 00:01:26.602
well, what are we
multiplying by every time

00:01:26.602 --> 00:01:28.622
we increase x by one?

00:01:28.622 --> 00:01:32.205
So three times our
common ratio two, to the

00:01:33.342 --> 00:01:35.675
to the x, to the x power.

00:01:37.272 --> 00:01:38.123
And you can verify that.

00:01:38.123 --> 00:01:39.298
Pick any of these.

00:01:39.298 --> 00:01:42.316
When x is equal to two,
it's gonna be three

00:01:42.316 --> 00:01:45.155
times two squared, which
is three times four,

00:01:45.155 --> 00:01:47.352
which is indeed equal to 12.

00:01:47.352 --> 00:01:50.053
And we can see that on a graph.

00:01:50.053 --> 00:01:54.036
So let me draw a quick
graph right over here.

00:01:54.036 --> 00:01:56.794
So, I'm having trouble
drawing a straight line.

00:01:56.794 --> 00:01:58.711
All right, there we go.

00:02:00.391 --> 00:02:01.385
And let's see.

00:02:01.385 --> 00:02:02.784
We could go, and they're gonna be

00:02:02.784 --> 00:02:06.121
on a slightly different
scale, my x and y axes.

00:02:06.121 --> 00:02:08.538
So this is x axis, y axis.

00:02:09.424 --> 00:02:12.924
And we go from negative one to one to two.

00:02:17.360 --> 00:02:19.379
Let's see, we're going
all the way up to 12.

00:02:19.379 --> 00:02:22.629
So let's see, this is three, six, nine,

00:02:26.550 --> 00:02:28.157
and let's say this is 12.

00:02:28.157 --> 00:02:30.358
We could just plot these points here.

00:02:30.358 --> 00:02:34.526
When x is negative one, y is 3/2.

00:02:34.526 --> 00:02:38.524
So looks like that, then at y equals zero,

00:02:38.524 --> 00:02:41.524
x is, when x is zero, y is three.

00:02:42.375 --> 00:02:45.369
When x equals one, y has doubled.

00:02:45.369 --> 00:02:46.548
It's now at six.

00:02:46.548 --> 00:02:49.242
When x is equal to two, y is 12.

00:02:49.242 --> 00:02:52.377
And you will see this tell-tale curve.

00:02:52.377 --> 00:02:54.861
And so there's a couple
of key features that we've

00:02:54.861 --> 00:02:56.844
Well, we've already talked
about several of them,

00:02:56.844 --> 00:02:59.317
but if you go to increasingly
negative x values,

00:02:59.317 --> 00:03:03.329
you will asymptote towards the x axis.

00:03:03.329 --> 00:03:04.768
It'll never quite get to zero

00:03:04.768 --> 00:03:06.268
as you get to more and
more negative values,

00:03:06.268 --> 00:03:07.655
but it'll definitely approach it.

00:03:07.655 --> 00:03:09.275
And as you get to more
and more positive values,

00:03:09.275 --> 00:03:10.790
it just kind of skyrockets up.

00:03:10.790 --> 00:03:12.630
We always, we've talked
about in previous videos

00:03:12.630 --> 00:03:15.770
how this will pass up any linear function

00:03:15.770 --> 00:03:18.579
or any linear graph eventually.

00:03:18.579 --> 00:03:22.329
Now, let's compare that
to exponential decay.

00:03:23.596 --> 00:03:26.179
Exponential, exponential decay.

00:03:29.575 --> 00:03:31.065
An easy way to think about it,

00:03:31.065 --> 00:03:33.452
instead of growing every
time you're increasing x,

00:03:33.452 --> 00:03:35.914
you're going to shrink
by a certain amount.

00:03:35.914 --> 00:03:37.711
You are going to decay.

00:03:37.711 --> 00:03:40.544
So let's set up another table here

00:03:41.474 --> 00:03:43.570
with x and y values.

00:03:43.570 --> 00:03:45.178
That was really a very,

00:03:45.178 --> 00:03:46.855
this is supposed to, when I press shift,

00:03:46.855 --> 00:03:50.122
it should create a straight
line but my computer,

00:03:50.122 --> 00:03:51.393
I've been eating next to my computer.

00:03:51.393 --> 00:03:53.579
Maybe there's crumbs in
the keyboard or something.

00:03:53.579 --> 00:03:54.725
(laughs) All right.

00:03:54.725 --> 00:03:55.570
So here we go.

00:03:55.570 --> 00:03:57.799
We have x and we have y.

00:03:57.799 --> 00:03:59.127
And so let's start with,

00:03:59.127 --> 00:04:01.124
let's say we start in the same place.

00:04:01.124 --> 00:04:03.541
So when x is zero, y is 3.

00:04:05.687 --> 00:04:09.484
But instead of doubling every
time we increase x by one,

00:04:09.484 --> 00:04:13.081
let's go by half every
time we increase x by one.

00:04:13.081 --> 00:04:15.568
So when x is equal to
one, we're gonna multiply

00:04:15.568 --> 00:04:19.306
by 1/2, and so we're gonna get to 3/2.

00:04:19.306 --> 00:04:24.014
Then when x is equal to two,
we'll multiply by 1/2 again

00:04:24.014 --> 00:04:27.597
and so we're going to get to 3/4

00:04:30.069 --> 00:04:31.430
and so on and so forth.

00:04:31.430 --> 00:04:32.795
And if we were to go to negative values,

00:04:32.795 --> 00:04:34.886
when x is equal to negative one,

00:04:34.886 --> 00:04:37.615
well, to go, if we're going
backwards in x by one,

00:04:37.615 --> 00:04:39.751
we would divide by 1/2,

00:04:39.751 --> 00:04:41.539
and so we would get to six.

00:04:41.539 --> 00:04:42.885
Or going from negative one to zero,

00:04:42.885 --> 00:04:46.642
as we increase x by one,
once again, we're multiplying

00:04:46.642 --> 00:04:49.253
we're multiplying by 1/2.

00:04:49.253 --> 00:04:51.097
And so how would we write
this as an equation?

00:04:51.097 --> 00:04:52.074
I encourage you to pause the video

00:04:52.074 --> 00:04:54.768
and see if you can write
it in a similar way.

00:04:54.768 --> 00:04:57.828
Well, it's gonna look something like this.

00:04:57.828 --> 00:05:00.244
It's gonna be y is equal to

00:05:00.244 --> 00:05:04.229
You have your, you could
have your y intercept here,

00:05:04.229 --> 00:05:06.240
the value of y when x is equal to zero,

00:05:06.240 --> 00:05:10.002
so it's three times, what's
our common ratio now?

00:05:10.002 --> 00:05:11.696
Well, every time we increase x by one,

00:05:11.696 --> 00:05:14.205
we're multiplying by 1/2

00:05:14.205 --> 00:05:18.372
so 1/2 and we're gonna
raise that to the x power.

00:05:22.105 --> 00:05:25.031
And so notice, these
are both exponentials.

00:05:25.031 --> 00:05:28.172
We have some, you could say
y intercept or initial value,

00:05:28.172 --> 00:05:30.323
it is being multiplied
by some common ratio

00:05:30.323 --> 00:05:32.322
to the power x.

00:05:32.322 --> 00:05:34.116
Some common ratio to the power x.

00:05:34.116 --> 00:05:37.582
But notice when you're
growing our common ratio

00:05:37.582 --> 00:05:40.159
and it actually turns
out to be a general idea,

00:05:40.159 --> 00:05:42.133
when you're growing, your common ratio,

00:05:42.133 --> 00:05:44.495
the absolute value of your common ratio

00:05:44.495 --> 00:05:46.624
is going to be greater than one.

00:05:46.624 --> 00:05:47.856
Let me write it down.

00:05:47.856 --> 00:05:49.534
So the absolute value of two in this case

00:05:49.534 --> 00:05:50.490
is greater than one.

00:05:50.490 --> 00:05:53.847
But when you're shrinking,
the absolute value of it

00:05:53.847 --> 00:05:56.094
is less than one.

00:05:56.094 --> 00:05:57.782
And that makes sense, because if the,

00:05:57.782 --> 00:05:59.507
if you have something
where the absolute value

00:05:59.507 --> 00:06:02.915
is less than one, like 1/2 or 3/4 or 0.9,

00:06:02.915 --> 00:06:04.562
every time you multiply it,

00:06:04.562 --> 00:06:07.924
you're gonna get a lower
and lower and lower value.

00:06:07.924 --> 00:06:10.147
And you could actually
see that in a graph.

00:06:10.147 --> 00:06:12.812
Let's graph the same
information right over here.

00:06:12.812 --> 00:06:14.546
And let me do it in a different color.

00:06:14.546 --> 00:06:17.125
I'll do it in a blue color.

00:06:17.125 --> 00:06:19.082
So when x is equal to negative one,

00:06:19.082 --> 00:06:21.367
y is equal to six.

00:06:21.367 --> 00:06:24.994
When x is equal to zero,
y is equal to three.

00:06:24.994 --> 00:06:27.966
When x is equal to one, y is equal to 3/2.

00:06:27.966 --> 00:06:30.990
When x is equal to two, y is equal to 3/4.

00:06:30.990 --> 00:06:33.385
And so on and so forth.

00:06:33.385 --> 00:06:37.293
And notice, because our common
ratios are the reciprocal

00:06:37.293 --> 00:06:39.216
of each other,

00:06:39.216 --> 00:06:42.919
that these two graphs look
like they've been flipped over,

00:06:42.919 --> 00:06:44.847
they look like they've
been flipped horizontally

00:06:44.847 --> 00:06:47.491
or flipped over the y axis.

00:06:47.491 --> 00:06:51.010
They're symmetric around that y axis.

00:06:51.010 --> 00:06:52.631
And what you will see in exponential decay

00:06:52.631 --> 00:06:54.889
is that things will get smaller
and smaller and smaller,

00:06:54.889 --> 00:06:57.587
but they'll never quite
exactly get to zero.

00:06:57.587 --> 00:06:58.969
It'll approach zero.

00:06:58.969 --> 00:07:01.333
It'll asymptote towards the x axis

00:07:01.333 --> 00:07:03.537
as x becomes more and more positive.

00:07:03.537 --> 00:07:05.494
Just as for exponential growth,

00:07:05.494 --> 00:07:07.856
if x becomes more and more negative,

00:07:07.856 --> 00:07:10.955
we asymptote towards the x axis.

00:07:10.955 --> 00:07:12.014
So that's the introduction.

00:07:12.014 --> 00:07:15.520
I'd use a very specific
example, but in general,

00:07:15.520 --> 00:07:18.903
if you have an equation of
the form y is equal to A

00:07:18.903 --> 00:07:21.819
times some common ratio to the x power

00:07:21.819 --> 00:07:23.607
We could write it like
that, just to make it

00:07:23.607 --> 00:07:24.640
a little bit clearer.

00:07:24.640 --> 00:07:27.213
There's a bunch of different
ways that we could write it.

00:07:27.213 --> 00:07:29.899
This is going to be exponential growth,

00:07:29.899 --> 00:07:33.231
so if the absolute value
of r is greater than one,

00:07:33.231 --> 00:07:35.774
then we're dealing with growth,

00:07:35.774 --> 00:07:37.631
because every time you multiply,

00:07:37.631 --> 00:07:39.534
every time you increase
x, you're multiplying

00:07:39.534 --> 00:07:42.142
by more and more r's is
one way to think about it.

00:07:42.142 --> 00:07:44.894
And if the absolute value
of r is less than one,

00:07:44.894 --> 00:07:46.920
you're dealing with decay.

00:07:46.920 --> 00:07:49.490
You're shrinking as x increases.

00:07:49.490 --> 00:07:51.418
And I'll let you think
about what happens when,

00:07:51.418 --> 00:07:54.046
what happens when r is equal to one?

00:07:54.046 --> 00:07:57.029
What are we dealing
with in that situation?

00:07:57.029 --> 00:07:58.309
And it's a bit of a trick question,

00:07:58.309 --> 00:07:59.236
because it's actually quite,

00:07:59.236 --> 00:08:00.434
oh, I'll just tell you.

00:08:00.434 --> 00:08:03.754
If r is equal to one, well
then, this thing right over here

00:08:03.754 --> 00:08:05.095
is always going to be equal to one

00:08:05.095 --> 00:08:07.742
and you boil down to just
the constant equation,

00:08:07.742 --> 00:08:10.912
y is equal to A, so this would
just be a horizontal line.


Comparing exponential and linear function

WEBVTT
Kind: captions
Language: en

00:00:00.463 --> 00:00:03.439
- [Instructor] Company
A is offering $10,000

00:00:03.439 --> 00:00:06.731
for the first month and
will increase the amount

00:00:06.731 --> 00:00:08.481
each month by $5,000.

00:00:10.677 --> 00:00:14.459
Company B is offering
$500 for the first month

00:00:14.459 --> 00:00:17.310
and will double their payment each month.

00:00:17.310 --> 00:00:21.091
For which monthly payment
will Company B's payment

00:00:21.091 --> 00:00:24.099
first exceed Company A's payment?

00:00:24.099 --> 00:00:26.905
So pause this video and
try to work that out.

00:00:26.905 --> 00:00:29.139
All right, let's work this out together.

00:00:29.139 --> 00:00:32.880
So let me set up a little bit of a table.

00:00:32.880 --> 00:00:34.797
So this is going to be,

00:00:36.707 --> 00:00:40.328
the first column I'm gonna have is month.

00:00:40.328 --> 00:00:42.495
The first column is month.

00:00:43.747 --> 00:00:48.022
The second column is how much
Company A is going to pay.

00:00:48.022 --> 00:00:50.592
And then third column,
let's think about how much

00:00:50.592 --> 00:00:53.240
Company B is going to pay.

00:00:53.240 --> 00:00:54.378
Well, they tell us a few things.

00:00:54.378 --> 00:00:58.735
They say Company A is offering
$10,000 for the first month.

00:00:58.735 --> 00:01:02.652
So in month one, Company
A is offering $10,000,

00:01:03.619 --> 00:01:06.240
we'll assume, well, I'll
just write the dollars there.

00:01:06.240 --> 00:01:10.228
And then Company B is offering
$500 for the first month.

00:01:10.228 --> 00:01:12.311
$500 for the first month.

00:01:13.562 --> 00:01:15.886
But then they tell us
Company A is offering,

00:01:15.886 --> 00:01:19.926
will increase the amount,
each month by $5,000.

00:01:19.926 --> 00:01:24.093
So month two will be 5,000
more, we'll get to $15,000.

00:01:25.469 --> 00:01:26.386
Month three

00:01:28.665 --> 00:01:30.415
we'll get to $20,000.

00:01:32.005 --> 00:01:34.255
Four, we'll get to $25,000.

00:01:35.570 --> 00:01:39.110
Five, I think you get the
point, we'll get to $30,000.

00:01:39.110 --> 00:01:41.277
Six, we'll get to $35,000.

00:01:42.822 --> 00:01:45.155
Seven, we'll get to $40,000.

00:01:46.769 --> 00:01:48.560
Let me scroll down a little bit.

00:01:48.560 --> 00:01:49.743
Month eight, I'll stop there.

00:01:49.743 --> 00:01:52.743
Month eight, we will get to $45,000.

00:01:54.432 --> 00:01:58.174
Let me extend these lines a little bit.

00:01:58.174 --> 00:02:00.372
Now let's think about what's
gonna happen with Company B.

00:02:00.372 --> 00:02:02.523
Company B is offering
500 for the first month,

00:02:02.523 --> 00:02:05.250
but will double their payment each month.

00:02:05.250 --> 00:02:07.729
So the second month is
gonna be double that.

00:02:07.729 --> 00:02:10.409
So that's going to be $1,000.

00:02:10.409 --> 00:02:13.550
Then we're gonna double
that again, $2,000.

00:02:13.550 --> 00:02:16.241
Then we're gonna double
that again, $4,000.

00:02:16.241 --> 00:02:19.091
Double that again, $16,000.

00:02:19.091 --> 00:02:21.341
Double that again, $32,000.

00:02:23.329 --> 00:02:25.607
Double that, oh, I skipped one.

00:02:25.607 --> 00:02:27.970
I went from 4,000 to 16,000.

00:02:27.970 --> 00:02:29.137
4,000, $8,000.

00:02:30.987 --> 00:02:34.199
Then we double it again, $16,000.

00:02:34.199 --> 00:02:38.367
Again, 32, I sound like
my two year old again.

00:02:38.367 --> 00:02:40.277
All right, $32,000.

00:02:40.277 --> 00:02:42.962
Then we get to $64,000.

00:02:42.962 --> 00:02:44.548
And at that point something
interesting happens.

00:02:44.548 --> 00:02:46.617
It's actually good that I
went to the eighth month

00:02:46.617 --> 00:02:48.491
because every month
before the eighth month,

00:02:48.491 --> 00:02:52.028
Company A's payment was higher
until that eighth month.

00:02:52.028 --> 00:02:56.716
In that eighth month, Company
B is going to pay more.

00:02:56.716 --> 00:02:59.281
So first we can just
answer their question.

00:02:59.281 --> 00:03:01.518
For which monthly payment
will Company B's payment

00:03:01.518 --> 00:03:04.083
first exceed Company A's payment?

00:03:04.083 --> 00:03:07.623
Well, that is month eight, month eight.

00:03:07.623 --> 00:03:10.843
And there's a broader
lesson going on here.

00:03:10.843 --> 00:03:14.626
You might recognize that the
rate at which Company A's

00:03:14.626 --> 00:03:17.147
payment is increasing is linear.

00:03:17.147 --> 00:03:19.994
Every month it increases
by the same amount.

00:03:19.994 --> 00:03:22.161
So plus 5,000, plus 5,000.

00:03:24.477 --> 00:03:27.199
It increases by 5,000 the same amount.

00:03:27.199 --> 00:03:30.084
Company B is increasing exponentially.

00:03:30.084 --> 00:03:32.449
It's increasing by the
same factor every time,

00:03:32.449 --> 00:03:34.650
so we're multiplying by
the same value every time.

00:03:34.650 --> 00:03:38.513
We're multiplying by two,
we're multiplying by two,

00:03:38.513 --> 00:03:40.096
multiplying by two.

00:03:41.080 --> 00:03:43.766
And so there's actually a
very interesting thing here

00:03:43.766 --> 00:03:45.967
that you can actually
make the general statement

00:03:45.967 --> 00:03:49.562
that an exponential function
will, we'll want something

00:03:49.562 --> 00:03:52.569
that is exponentially
increasing, will eventually

00:03:52.569 --> 00:03:56.759
always surpass something
that is linearly increasing.

00:03:56.759 --> 00:04:00.010
And it doesn't matter what
the initial situation is,

00:04:00.010 --> 00:04:02.169
and it also doesn't even matter that rate

00:04:02.169 --> 00:04:06.528
of exponential increase,
it will eventually always

00:04:06.528 --> 00:04:09.291
pass up something that's
increasing linearly.

00:04:09.291 --> 00:04:12.049
And you can think about
that visually if you like.

00:04:12.049 --> 00:04:16.444
If I were to draw a visual
function, a linear function,

00:04:16.444 --> 00:04:20.074
so this is x-axis, this is the y-axis,

00:04:20.074 --> 00:04:22.274
a linear function, well, this is gonna be

00:04:22.274 --> 00:04:23.861
described by a line.

00:04:23.861 --> 00:04:26.944
So it could look something like this.

00:04:27.848 --> 00:04:31.549
A linear function is always
gonna be a line of some slope.

00:04:31.549 --> 00:04:34.116
And an exponential function,
even though it might

00:04:34.116 --> 00:04:37.616
start a little bit slower, it's eventually

00:04:38.802 --> 00:04:41.651
going to pass up the linear function.

00:04:41.651 --> 00:04:43.769
And this is going to be the
case even if the linear function

00:04:43.769 --> 00:04:46.666
has a pretty high slope or a
pretty high starting point,

00:04:46.666 --> 00:04:49.011
if it's something like that,
and even if the exponential

00:04:49.011 --> 00:04:52.797
function is starting pretty
slow, it will eventually,

00:04:52.797 --> 00:04:55.686
and even if it's compounding
or growing relatively slow

00:04:55.686 --> 00:04:58.826
but exponentially, you know
if it's going 2% or 3%,

00:04:58.826 --> 00:05:02.404
it still will eventually
pass up the linear function

00:05:02.404 --> 00:05:04.237
which is pretty cool.


Identifying graph for exponential

WEBVTT
Kind: captions
Language: en

00:00:00.344 --> 00:00:02.629
- [Instructor] Alright,
we are asked to choose

00:00:02.629 --> 00:00:04.707
the graph of the function.

00:00:04.707 --> 00:00:06.423
And the function is f(x) is equal to

00:00:06.423 --> 00:00:08.305
two, times three to the x

00:00:08.305 --> 00:00:10.099
and we have three choices here.

00:00:10.099 --> 00:00:12.220
So, pause this video and
see if you can determine

00:00:12.220 --> 00:00:14.176
which of these three graphs

00:00:14.176 --> 00:00:17.893
actually is the graph of f(x).

00:00:17.893 --> 00:00:19.975
Let's work through this together.

00:00:19.975 --> 00:00:21.970
So, whenever I have a function like this,

00:00:21.970 --> 00:00:23.886
which is an exponential function,

00:00:23.886 --> 00:00:25.478
because I'm taking some number

00:00:25.478 --> 00:00:30.172
and I'm multiplying it by some
other number to some power.

00:00:30.172 --> 00:00:32.702
So, that tells me that I'm
dealing with an exponential.

00:00:32.702 --> 00:00:34.783
So, I like to think about two things.

00:00:34.783 --> 00:00:37.890
What happens when x equals zero?

00:00:37.890 --> 00:00:41.069
What is the value of our function?

00:00:41.069 --> 00:00:42.865
Well, when you just look at this function,

00:00:42.865 --> 00:00:46.454
this would be two,
times three to the zero.

00:00:46.454 --> 00:00:48.984
Which is equal to, three
to the zero is one.

00:00:48.984 --> 00:00:50.780
It's equal to two.

00:00:50.780 --> 00:00:52.206
So, one way to think about it.

00:00:52.206 --> 00:00:55.664
In the graph of y is equal to f(x),

00:00:55.664 --> 00:00:59.908
when x is equal to
zero, y is equal to two.

00:00:59.908 --> 00:01:01.456
Or another way to think about it is

00:01:01.456 --> 00:01:03.490
this value in exponential function,

00:01:03.490 --> 00:01:04.959
sometimes called the initial value,

00:01:04.959 --> 00:01:06.753
if you were thinking of the x-axis.

00:01:06.753 --> 00:01:09.243
Instead of the x-axis, you're
thinking about the time axis

00:01:09.243 --> 00:01:10.547
or the t-axis.

00:01:10.547 --> 00:01:12.630
That's why it's sometimes
called the initial value.

00:01:12.630 --> 00:01:15.154
But the y-intercept is
gonna be described by that

00:01:15.154 --> 00:01:17.387
when you have a function of this form.

00:01:17.387 --> 00:01:19.501
And you saw it right over there, f(0).

00:01:19.501 --> 00:01:20.606
Three to the zero's one.

00:01:20.606 --> 00:01:22.116
You're just left with the two.

00:01:22.116 --> 00:01:25.180
So, which of these have
a y-intercept of two?

00:01:25.180 --> 00:01:27.505
Well, here, the
y-intercept looks like one.

00:01:27.505 --> 00:01:29.710
Here, the y-intercept looks like three.

00:01:29.710 --> 00:01:32.000
Here, the y-intercept is two.

00:01:32.000 --> 00:01:34.418
So, just through elimination
through that alone,

00:01:34.418 --> 00:01:36.128
we can feel pretty good that

00:01:36.128 --> 00:01:38.941
this third graph is probably the choice.

00:01:38.941 --> 00:01:40.570
But let's keep on analyzing it

00:01:40.570 --> 00:01:41.670
to feel even better about it.

00:01:41.670 --> 00:01:43.711
And so, we have the skills for really any

00:01:43.711 --> 00:01:46.974
exponential function
that we might run into.

00:01:46.974 --> 00:01:48.808
Well, the other thing to realize.

00:01:48.808 --> 00:01:51.660
This number, three, is often referred to

00:01:51.660 --> 00:01:53.160
as a common ratio.

00:01:54.145 --> 00:01:57.856
And that's because every
time you increase x by one,

00:01:57.856 --> 00:02:00.469
you're gonna be taking
three to a one higher power.

00:02:00.469 --> 00:02:03.446
Or you're essentially gonna
be multiplying by three again.

00:02:03.446 --> 00:02:07.229
So, for example, f(1) is going to be

00:02:07.229 --> 00:02:09.917
equal to two, times three to the one.

00:02:09.917 --> 00:02:12.733
Two, times three to the
one or two times three,

00:02:12.733 --> 00:02:14.039
which is equal to six.

00:02:14.039 --> 00:02:16.851
So, from f(0) to f(1), you essentially

00:02:16.851 --> 00:02:18.687
have to multiply by three.

00:02:18.687 --> 00:02:20.443
And you keep multiplying by three.

00:02:20.443 --> 00:02:23.866
f(2) you're gonna multiply by three again.

00:02:23.866 --> 00:02:27.004
It's gonna be two, times three squared,

00:02:27.004 --> 00:02:28.471
which is equal to 18.

00:02:28.471 --> 00:02:32.467
And so, once again, when
I increased my x by one,

00:02:32.467 --> 00:02:36.511
I'm multiplying the value
of my function by three.

00:02:36.511 --> 00:02:38.056
So, let's just see which of these do this.

00:02:38.056 --> 00:02:40.548
This one we said has
the wrong y-intercept,

00:02:40.548 --> 00:02:44.590
but, as we go from x equals
zero to x equals one,

00:02:44.590 --> 00:02:46.624
we are going from one to three.

00:02:46.624 --> 00:02:50.371
And then, we are going from three till

00:02:50.371 --> 00:02:52.163
looks like pretty close to nine.

00:02:52.163 --> 00:02:53.713
So, it does look like this does

00:02:53.713 --> 00:02:55.060
have a common ratio of three.

00:02:55.060 --> 00:02:57.959
It just does have a different y-intercept

00:02:57.959 --> 00:02:59.225
than the function we care about.

00:02:59.225 --> 00:03:00.327
This looks like the graph

00:03:00.327 --> 00:03:03.833
f(x) is equal to just
one, times 3 to the x.

00:03:03.833 --> 00:03:06.315
Here, we're starting at three.

00:03:06.315 --> 00:03:08.395
And then, when x equals one,

00:03:08.395 --> 00:03:12.562
it looks like we are doubling
every time x increases by one.

00:03:14.220 --> 00:03:18.139
So, this looks like the
graph of y is equal to...

00:03:18.139 --> 00:03:20.461
I have what we could
call our initial value,

00:03:20.461 --> 00:03:22.868
our y-intercept, three.

00:03:22.868 --> 00:03:25.435
And, if we're doubling every
time, we increase by one.

00:03:25.435 --> 00:03:27.516
Three, times two to the x.

00:03:27.516 --> 00:03:28.820
That's this graph here.

00:03:28.820 --> 00:03:30.207
As I said, this first graph looks like

00:03:30.207 --> 00:03:33.540
y is equal to one, times three to the x.

00:03:35.105 --> 00:03:36.450
We are tripling every time.

00:03:36.450 --> 00:03:37.837
One, times three to the x.

00:03:37.837 --> 00:03:41.309
Or we could just say y is
equal to three to the x.

00:03:41.309 --> 00:03:42.531
Now, this one here better work,

00:03:42.531 --> 00:03:44.854
'cause we already picked
it as our solution.

00:03:44.854 --> 00:03:47.015
So, let's see if that's actually the case.

00:03:47.015 --> 00:03:50.890
So, as we increase by one,
we should multiply by three.

00:03:50.890 --> 00:03:54.693
So, two times three is, indeed, six.

00:03:54.693 --> 00:03:56.407
And then, when you
increase by another one,

00:03:56.407 --> 00:03:57.875
we should go to 18.

00:03:57.875 --> 00:03:59.591
And that's kind of off the charts here,

00:03:59.591 --> 00:04:01.017
but it does seem reasonable to see that

00:04:01.017 --> 00:04:04.487
we are multiplying by three every time.

00:04:04.487 --> 00:04:06.889
And you could also go the other way.

00:04:06.889 --> 00:04:09.538
If you're going down by one,

00:04:09.538 --> 00:04:11.334
you should be dividing by three.

00:04:11.334 --> 00:04:12.718
So, two divided by three,

00:04:12.718 --> 00:04:16.512
this does look pretty close to 2/3.

00:04:16.512 --> 00:04:20.679
So, we should feel very
good about our third choice.


Interpretting exponential expression

WEBVTT
Kind: captions
Language: en

00:00:00.543 --> 00:00:03.845
- [Instructor] The expression
five times two to the T

00:00:03.845 --> 00:00:06.419
gives the number of leaves in a plant

00:00:06.419 --> 00:00:10.180
as a function of the number
of weeks since it was planted.

00:00:10.180 --> 00:00:12.879
What does two represent
in this expression?

00:00:12.879 --> 00:00:14.467
So pause this video and see
if you can figure it out

00:00:14.467 --> 00:00:15.467
on your own.

00:00:16.647 --> 00:00:20.171
All right, so let's look at
the expression right over here.

00:00:20.171 --> 00:00:22.307
We could write it as defining a function.

00:00:22.307 --> 00:00:26.043
So we could say leaves
as a function of time

00:00:26.043 --> 00:00:29.460
is equal to five times two to the T power

00:00:31.559 --> 00:00:33.171
and so we could try this out a little bit.

00:00:33.171 --> 00:00:35.521
If we say well what is L of zero?

00:00:35.521 --> 00:00:37.061
That would be T equals zero.

00:00:37.061 --> 00:00:39.287
That's when we are at zero
weeks after it was planted.

00:00:39.287 --> 00:00:41.346
So this is right when it was planted.

00:00:41.346 --> 00:00:44.596
Well that's five times two to the zero,

00:00:45.495 --> 00:00:47.358
which is just two to the zero's just one.

00:00:47.358 --> 00:00:48.853
So it's equal to five

00:00:48.853 --> 00:00:51.780
and so when you see an
exponential expression

00:00:51.780 --> 00:00:53.557
or an exponential function like this,

00:00:53.557 --> 00:00:55.931
that is why this number out here

00:00:55.931 --> 00:00:59.843
is often referred to
as your initial value.

00:00:59.843 --> 00:01:01.296
Initial.

00:01:01.296 --> 00:01:02.463
Initial value.

00:01:04.355 --> 00:01:06.846
And so let's explore
this a little bit more.

00:01:06.846 --> 00:01:08.235
What is L of one?

00:01:08.235 --> 00:01:09.990
What happens after one week?

00:01:09.990 --> 00:01:13.386
Well that's gonna be five
times two to the first power

00:01:13.386 --> 00:01:15.298
or five times two.

00:01:15.298 --> 00:01:19.117
So going from when it was
planted to the first week,

00:01:19.117 --> 00:01:20.543
we are multiplying by two.

00:01:20.543 --> 00:01:22.338
The number of leaves doubles.

00:01:22.338 --> 00:01:24.339
Well what happens after two weeks,

00:01:24.339 --> 00:01:25.514
the number of leaves after two weeks?

00:01:25.514 --> 00:01:30.112
Well that's gonna be five
times two to the second power.

00:01:30.112 --> 00:01:32.995
Well that's the number that
you had in the first week

00:01:32.995 --> 00:01:34.154
times two.

00:01:34.154 --> 00:01:38.176
So it looks like every
week we are doubling,

00:01:38.176 --> 00:01:40.584
we are multiplying by two

00:01:40.584 --> 00:01:44.103
and that's why this
number right over here,

00:01:44.103 --> 00:01:46.290
which is what the question
is about, the two,

00:01:46.290 --> 00:01:50.380
this is often referred
to as the common ratio.

00:01:50.380 --> 00:01:52.184
Common ratio.

00:01:52.184 --> 00:01:54.991
Because between any two successive weeks,

00:01:54.991 --> 00:01:58.698
the ratio between say week
two and week one is two.

00:01:58.698 --> 00:02:00.698
Week two is double week one.

00:02:00.698 --> 00:02:03.701
Week one is double week zero.

00:02:03.701 --> 00:02:05.256
So let's see which of these choices

00:02:05.256 --> 00:02:07.284
actually match up to that.

00:02:07.284 --> 00:02:09.958
There were initially
two leaves in the plant.

00:02:09.958 --> 00:02:11.519
Well we know that there weren't
two leaves on the plant.

00:02:11.519 --> 00:02:13.580
Our initial value was five.

00:02:13.580 --> 00:02:15.997
So let me cross that one out.

00:02:16.938 --> 00:02:19.881
The number of leaves is
multiplied by two each week.

00:02:19.881 --> 00:02:21.700
Well that's exactly
what we just described.

00:02:21.700 --> 00:02:23.704
So I like that choice.

00:02:23.704 --> 00:02:25.979
Let's look at the last
one just for good measure.

00:02:25.979 --> 00:02:28.813
The plant was planted two weeks ago.

00:02:28.813 --> 00:02:31.282
Well no, they don't tell
us anything about that.

00:02:31.282 --> 00:02:34.257
This is a general expression

00:02:34.257 --> 00:02:37.253
for T weeks after it was planted.

00:02:37.253 --> 00:02:39.409
So they're not saying when
it was actually planted.

00:02:39.409 --> 00:02:40.415
So we could rule that out

00:02:40.415 --> 00:02:43.906
and we feel good about that second choice.


Modeling with basic exponential function

WEBVTT
Kind: captions
Language: en

00:00:00.331 --> 00:00:03.891
- [Instructor] There are
170 deer on a reservation.

00:00:03.891 --> 00:00:06.851
The deer population is increasing

00:00:06.851 --> 00:00:09.434
at a rate of 30% per year.

00:00:10.453 --> 00:00:14.873
Write a function that gives
the deer population P of t

00:00:14.873 --> 00:00:17.873
on the reservation t years from now.

00:00:19.424 --> 00:00:20.411
All right, let's think about this.

00:00:20.411 --> 00:00:21.684
And like always, pause this video,

00:00:21.684 --> 00:00:23.706
and see if you can work
it out on your own.

00:00:23.706 --> 00:00:26.283
Well, let's think about what P of zero is.

00:00:26.283 --> 00:00:27.644
P of zero, this is going to be

00:00:27.644 --> 00:00:29.408
the initial population of deer,

00:00:29.408 --> 00:00:31.822
the population at time zero.

00:00:31.822 --> 00:00:35.345
Well, we know that, that's
going to be the 170 deer

00:00:35.345 --> 00:00:38.081
that we start on the reservation.

00:00:38.081 --> 00:00:40.033
Now, let's think about what P of one is.

00:00:40.033 --> 00:00:43.332
What's going to be the
population after one year?

00:00:43.332 --> 00:00:46.550
Well, it's going to be our
original population, 170,

00:00:46.550 --> 00:00:49.633
but then it increases at
a rate of 30% per year.

00:00:49.633 --> 00:00:54.614
So it's going to be 170
plus another 30% of 170.

00:00:54.614 --> 00:00:58.495
So I could write that as 30% times 170.

00:00:58.495 --> 00:01:01.407
Or I could write this as 170

00:01:01.407 --> 00:01:02.240
plus 0.3

00:01:03.307 --> 00:01:04.140
times 170.

00:01:05.240 --> 00:01:09.768
30% as a decimal is the same
thing as 30/100 or 3/10.

00:01:09.768 --> 00:01:12.242
Or I could write this as,

00:01:12.242 --> 00:01:14.522
if I factor out 170,

00:01:14.522 --> 00:01:16.573
I would get 170

00:01:16.573 --> 00:01:18.061
times

00:01:18.061 --> 00:01:18.894
one plus

00:01:20.031 --> 00:01:22.948
0.3, which is the same thing as 170

00:01:24.019 --> 00:01:24.936
times 1.03.

00:01:27.327 --> 00:01:29.971
And this is a really good
thing to take a hard look at

00:01:29.971 --> 00:01:31.228
'cause you'll see it a lot

00:01:31.228 --> 00:01:32.983
when we're growing by a certain rate,

00:01:32.983 --> 00:01:34.263
when we're dealing with

00:01:34.263 --> 00:01:37.821
what turns out to be
exponential functions.

00:01:37.821 --> 00:01:39.374
If we are growing,

00:01:39.374 --> 00:01:43.609
well, I almost made a
mistake there, it's 1.3.

00:01:43.609 --> 00:01:45.334
So here you go, 1.3.

00:01:45.334 --> 00:01:47.001
One plus 0.3 is 1.3.

00:01:48.457 --> 00:01:51.553
So once again, take a hard
look at this right over here

00:01:51.553 --> 00:01:52.924
because it's going to be something

00:01:52.924 --> 00:01:55.839
that you see a lot with
exponential functions.

00:01:55.839 --> 00:01:58.006
When you grow by 30%,

00:01:59.428 --> 00:02:03.145
that means you keep your
100% that you had before,

00:02:03.145 --> 00:02:05.750
and then you add another 30%.

00:02:05.750 --> 00:02:09.917
And so you would multiply your
original quantity by 130%,

00:02:11.436 --> 00:02:14.971
and 130% is the same thing as 1.3.

00:02:14.971 --> 00:02:18.336
So if you are growing by 30%,

00:02:18.336 --> 00:02:21.248
you are growing by 3/10,

00:02:21.248 --> 00:02:25.204
you would multiply your
initial quantity by 1.3.

00:02:25.204 --> 00:02:27.447
So let's use that idea to keep going.

00:02:27.447 --> 00:02:31.019
So what is the population after two years?

00:02:31.019 --> 00:02:33.480
Well, you would start that second year

00:02:33.480 --> 00:02:35.347
with the population at
the end of one year.

00:02:35.347 --> 00:02:38.447
So it's going to be that 170

00:02:38.447 --> 00:02:39.280
times 1.3.

00:02:40.300 --> 00:02:42.353
And then, over that year,
you're going to grow

00:02:42.353 --> 00:02:44.120
by another 30%.

00:02:44.120 --> 00:02:46.156
So if you're gonna grow by another 30%,

00:02:46.156 --> 00:02:49.825
that's equivalent to
multiplying by 1.3 again.

00:02:49.825 --> 00:02:53.992
Or you could say that this
is equal to 170 times 1.3

00:02:54.976 --> 00:02:56.643
to the second power.

00:02:57.492 --> 00:02:59.515
And so I think you see
where this is going.

00:02:59.515 --> 00:03:02.466
If we wanted to write a general P of t,

00:03:02.466 --> 00:03:06.104
so if we just want to
write a general P of t,

00:03:06.104 --> 00:03:10.086
it's going to be whatever
we started with, 170,

00:03:10.086 --> 00:03:13.366
and we're going to multiply that by 1.3

00:03:13.366 --> 00:03:17.669
however many times, however
many years have gone by.

00:03:17.669 --> 00:03:19.554
So to the t power

00:03:19.554 --> 00:03:22.833
because, for every year, we grow by 30%,

00:03:22.833 --> 00:03:26.546
which is equivalent mathematically
to multiplying by 1.3.

00:03:26.546 --> 00:03:28.679
So after 100 years,

00:03:28.679 --> 00:03:32.429
it would be 170 times
1.3 to the 100th power.


Expression for compound or exponential growth

WEBVTT
Kind: captions
Language: en

00:00:00.489 --> 00:00:04.131
- [Instructor] You put
$3,800 in a savings account.

00:00:04.131 --> 00:00:08.225
The bank will provide
1.8% interest on the money

00:00:08.225 --> 00:00:09.173
in the account every year.

00:00:09.173 --> 00:00:10.185
Another way of saying that,

00:00:10.185 --> 00:00:14.352
the money in the savings account
will grow by 1.8% per year.

00:00:15.872 --> 00:00:19.048
Write an expression that
describes how much money

00:00:19.048 --> 00:00:22.024
will be in the account in 15 years.

00:00:22.024 --> 00:00:23.323
So let's just think
about this a little bit.

00:00:23.323 --> 00:00:25.795
Let's just think about
the starting amount.

00:00:25.795 --> 00:00:29.023
So in the start, we're
just gonna put $3,800.

00:00:29.023 --> 00:00:31.690
We could view that as year zero.

00:00:33.638 --> 00:00:35.089
Actually, let me write it that way.

00:00:35.089 --> 00:00:38.672
So the start is the
same thing as year zero

00:00:40.959 --> 00:00:43.792
and we're gonna start with $3,800.

00:00:45.478 --> 00:00:47.473
Now let's think about year one.

00:00:47.473 --> 00:00:50.242
How much money will we
have after one year?

00:00:50.242 --> 00:00:54.409
Well, we would have the original
amount that we put, $3,800,

00:00:55.958 --> 00:00:58.005
and then we're gonna get the
amount that we get an interest

00:00:58.005 --> 00:01:01.122
and they say that the bank
will provide 1.8% interest

00:01:01.122 --> 00:01:02.789
on the money in the account

00:01:02.789 --> 00:01:06.039
so it'll be plus 1.8% times $3,800

00:01:14.735 --> 00:01:16.602
and we could also write this as a decimal.

00:01:16.602 --> 00:01:20.767
This is equal to 3,800
plus and I'll just write,

00:01:20.767 --> 00:01:22.429
I'll switch the order
of multiplication here,

00:01:22.429 --> 00:01:24.346
plus 3,800 times 0.018.

00:01:28.553 --> 00:01:32.720
1.8% is the same thing
as 18,000ths or 1.800ths

00:01:35.085 --> 00:01:37.985
depending on how you want to pronounce it.

00:01:37.985 --> 00:01:38.939
And so here you might say,

00:01:38.939 --> 00:01:41.751
"Well, there's an interesting
potential simplification

00:01:41.751 --> 00:01:42.954
"mathematically here."

00:01:42.954 --> 00:01:46.566
I could factor 3,800 out
of each of these terms.

00:01:46.566 --> 00:01:49.690
I have a 3,800 here, I have a 3,800 here

00:01:49.690 --> 00:01:51.529
so why don't I factor it out?

00:01:51.529 --> 00:01:53.456
Essentially undistributed.

00:01:53.456 --> 00:01:56.373
So this is going to be 3,800 times,

00:01:57.986 --> 00:01:58.819
when you factor it out here,

00:01:58.819 --> 00:02:00.402
you get a one plus,

00:02:01.826 --> 00:02:03.006
when you factor it out here,

00:02:03.006 --> 00:02:04.089
you get 0.018

00:02:05.929 --> 00:02:10.096
and so I could just rewrite
this as 3,800 times 1.018.

00:02:16.185 --> 00:02:18.329
So this is an interesting time to pause.

00:02:18.329 --> 00:02:20.396
We're not at the full answer yet,

00:02:20.396 --> 00:02:22.245
how much will we have in 15 years,

00:02:22.245 --> 00:02:23.794
but we have an interesting expression

00:02:23.794 --> 00:02:26.482
for how much we have after one year.

00:02:26.482 --> 00:02:30.557
Notice that if the
money is growing by 1.8%

00:02:30.557 --> 00:02:33.807
or another way it was growing by 0.018,

00:02:35.114 --> 00:02:37.256
that's equivalent to multiplying

00:02:37.256 --> 00:02:39.641
the amount that we started the year with

00:02:39.641 --> 00:02:43.808
by one plus the amount that
it's growing by or 1.018.

00:02:45.462 --> 00:02:47.577
And once again, why does
this make intuitive sense?

00:02:47.577 --> 00:02:49.598
Because at the end of the year,

00:02:49.598 --> 00:02:52.388
you're going to have the
original amount that you put,

00:02:52.388 --> 00:02:54.131
that's what that one really represents,

00:02:54.131 --> 00:02:57.061
and then plus you're gonna have
the amount that you grew by

00:02:57.061 --> 00:02:59.120
so you multiply both the sum here

00:02:59.120 --> 00:03:00.740
times the original amount you put

00:03:00.740 --> 00:03:04.498
and that's how much you'll
have at the end of year one.

00:03:04.498 --> 00:03:06.662
What about year two?

00:03:06.662 --> 00:03:07.662
So year two.

00:03:11.047 --> 00:03:13.790
Well, we know what we're going
to start with in year two.

00:03:13.790 --> 00:03:16.548
We're gonna start with whatever
we finished year one with.

00:03:16.548 --> 00:03:20.215
So we're gonna start
with 3,800 times 1.018,

00:03:23.034 --> 00:03:27.201
but then it's gonna grow
by 1.8% or grow by 0.018.

00:03:29.983 --> 00:03:32.476
And we already said, if you're
gonna grow by that amount,

00:03:32.476 --> 00:03:36.226
that's equivalent to
multiplying it by 1.018.

00:03:38.290 --> 00:03:42.373
Well, this is the same
thing as 3,800 times 1.018

00:03:44.867 --> 00:03:46.898
to the second power.

00:03:46.898 --> 00:03:49.475
I think you see where this is going.

00:03:49.475 --> 00:03:52.142
Every time we grow by 1.8%,

00:03:52.976 --> 00:03:56.024
we're gonna multiply by 1.018.

00:03:56.024 --> 00:03:58.194
And if we're thinking about
15 years in the future,

00:03:58.194 --> 00:04:00.019
we're gonna do that 15 times.

00:04:00.019 --> 00:04:01.132
So one year in the future,

00:04:01.132 --> 00:04:02.972
your exponent here is essentially one.

00:04:02.972 --> 00:04:04.831
Two years, your exponent is two.

00:04:04.831 --> 00:04:08.677
So year 15, I can just
cut to the chase here,

00:04:08.677 --> 00:04:09.594
so year 15,

00:04:12.351 --> 00:04:13.699
well that's just going to be,

00:04:13.699 --> 00:04:17.065
we're going to have the
original amount that we invested

00:04:17.065 --> 00:04:20.315
and we are going to grow 1.018 15 times

00:04:22.507 --> 00:04:25.659
so we're gonna multiply
by this amount 15 times

00:04:25.659 --> 00:04:27.812
to get the final amount.

00:04:27.812 --> 00:04:28.759
And one of the fun things,

00:04:28.759 --> 00:04:30.892
this is actually called compound growth

00:04:30.892 --> 00:04:33.413
where every year you
grow on top of the amount

00:04:33.413 --> 00:04:34.848
that you had before.

00:04:34.848 --> 00:04:36.967
You'll see if you type
this into a calculator

00:04:36.967 --> 00:04:41.077
that even though 1.8% per
year does not seem like a lot,

00:04:41.077 --> 00:04:43.962
over 15 years it actually would amount

00:04:43.962 --> 00:04:45.778
to a reasonable amount,

00:04:45.778 --> 00:04:47.098
but this is the expression.

00:04:47.098 --> 00:04:48.490
They're not asking us to calculate it.

00:04:48.490 --> 00:04:51.211
They just want us to know
an expression that describes

00:04:51.211 --> 00:04:55.378
how much money will be in
the account in 15 years.


5th roots

WEBVTT
Kind: captions
Language: en

00:00:00.421 --> 00:00:02.335
- [Instructor] Let's
see if we can calculate

00:00:02.335 --> 00:00:04.085
the fifth root of 32.

00:00:08.289 --> 00:00:10.412
So, like always, pause the
video and see if you can

00:00:10.412 --> 00:00:12.690
figure this out on your own.

00:00:12.690 --> 00:00:16.316
So, let's just remind
ourselves what a fifth root is.

00:00:16.316 --> 00:00:20.066
So, if x is equal to the fifth root of 32,

00:00:21.042 --> 00:00:22.551
that's the same thing as saying

00:00:22.551 --> 00:00:25.551
x to the fifth power is equal to 32.

00:00:27.612 --> 00:00:29.242
So, we have to find some number where,

00:00:29.242 --> 00:00:31.445
if you take five of them
and multiply them together,

00:00:31.445 --> 00:00:33.198
you'd get 32.

00:00:33.198 --> 00:00:35.400
So, there is a couple of
ways to approach this.

00:00:35.400 --> 00:00:36.749
Especially when you're dealing with these

00:00:36.749 --> 00:00:39.358
really high order roots here.

00:00:39.358 --> 00:00:43.108
So, let me rewrite the
fifth root of 32 here.

00:00:45.236 --> 00:00:47.805
One way is you could try to factor 32

00:00:47.805 --> 00:00:51.761
and see are there factors
that show up five times?

00:00:51.761 --> 00:00:54.655
So, 32 you might immediately
recognize is an even number.

00:00:54.655 --> 00:00:56.857
So, it's gonna be divisible by two.

00:00:56.857 --> 00:00:58.572
It's two times 16.

00:00:58.572 --> 00:01:00.405
16 is two times eight.

00:01:01.793 --> 00:01:03.793
Eight is two times four.

00:01:05.098 --> 00:01:07.300
Four is two times two.

00:01:07.300 --> 00:01:09.174
So, in this case, doing the

00:01:09.174 --> 00:01:10.881
factoring technique worked out well.

00:01:10.881 --> 00:01:12.310
'Cause we see that this is

00:01:12.310 --> 00:01:14.654
two times two times
two times two times two

00:01:14.654 --> 00:01:16.764
or two to the fifth power.

00:01:16.764 --> 00:01:19.257
You could rewrite this as the fifth root

00:01:19.257 --> 00:01:21.180
of two to the fifth power,

00:01:21.180 --> 00:01:24.198
which is, of course,
going to be equal to two.

00:01:24.198 --> 00:01:26.615
Two to the fifth power is 32.

00:01:29.584 --> 00:01:31.254
Now, let's do another one.

00:01:31.254 --> 00:01:34.763
It's gonna be a little bit harder.

00:01:34.763 --> 00:01:38.596
Let's say we wanna take
the fifth root of 243.

00:01:41.296 --> 00:01:43.747
So, now, a much, much larger number.

00:01:43.747 --> 00:01:45.261
So, there's a couple of ways to do this.

00:01:45.261 --> 00:01:47.262
One, you could try the factoring.

00:01:47.262 --> 00:01:48.363
Although, that's gonna be harder

00:01:48.363 --> 00:01:49.546
now that it's a larger number.

00:01:49.546 --> 00:01:52.034
Or you could do a little
bit of trial and error.

00:01:52.034 --> 00:01:54.969
Doing higher roots without the aid

00:01:54.969 --> 00:01:56.438
of some type of calculator or something

00:01:56.438 --> 00:01:59.536
is a little bit more complicated.

00:01:59.536 --> 00:02:02.921
So, here, if we wanted to
do the factoring technique.

00:02:02.921 --> 00:02:06.306
We could say, alright,
it's not divisible by two.

00:02:06.306 --> 00:02:08.557
I like to start with the
smallest possible factor.

00:02:08.557 --> 00:02:10.429
So, it's not divisible by two.

00:02:10.429 --> 00:02:12.466
Is it divisible by three?

00:02:12.466 --> 00:02:14.178
And you might be familiar with the test

00:02:14.178 --> 00:02:16.378
to see if something is divisible by three.

00:02:16.378 --> 00:02:18.253
You add up the digits and see if that

00:02:18.253 --> 00:02:20.822
sum of the digits is divisible by three.

00:02:20.822 --> 00:02:25.300
So, if I were to take
two plus four plus three,

00:02:25.300 --> 00:02:26.892
that is equal to nine.

00:02:26.892 --> 00:02:29.504
And so it is divisible by three.

00:02:29.504 --> 00:02:32.478
So, this is going to be
equal to three times...

00:02:32.478 --> 00:02:36.434
Let's see three goes into
240 80 times and then one.

00:02:36.434 --> 00:02:37.517
So, 81 times.

00:02:39.903 --> 00:02:42.392
And so, 81 is also divisible by three.

00:02:42.392 --> 00:02:45.123
I have a sense of where this is going now.

00:02:45.123 --> 00:02:49.810
It's three times 27,
which is three times nine.

00:02:49.810 --> 00:02:52.053
Which is three times three.

00:02:52.053 --> 00:02:53.403
So, using the factoring method,

00:02:53.403 --> 00:02:56.832
we're able to see that three
to the fifth power is 243.

00:02:56.832 --> 00:03:00.499
So, the fifth root of
243 is equal to three.

00:03:02.787 --> 00:03:04.335
Now, another way that
you could have done it

00:03:04.335 --> 00:03:06.213
is a little bit of trial and error.

00:03:06.213 --> 00:03:08.288
We already know..

00:03:08.288 --> 00:03:09.432
Well, we know that one to the fifth power

00:03:09.432 --> 00:03:10.938
is just going to be one.

00:03:10.938 --> 00:03:12.364
We know that two to the fifth power...

00:03:12.364 --> 00:03:13.382
We just calculated that.

00:03:13.382 --> 00:03:14.215
That's 32.

00:03:15.255 --> 00:03:17.254
Well, we now know what
three to the fifth is.

00:03:17.254 --> 00:03:18.191
Let's say we're just trying to

00:03:18.191 --> 00:03:20.068
zoom in on it a little bit.

00:03:20.068 --> 00:03:21.252
So, let's say, if you wanted to see

00:03:21.252 --> 00:03:23.496
what four to the fifth is.

00:03:23.496 --> 00:03:26.147
Well, that would be four times four

00:03:26.147 --> 00:03:28.583
times four times four times four.

00:03:28.583 --> 00:03:32.381
So, let's see, this is going to be 16.

00:03:32.381 --> 00:03:34.048
16 times four is 64.

00:03:37.029 --> 00:03:38.529
Times four is 256.

00:03:40.785 --> 00:03:42.585
And then, that times four...

00:03:42.585 --> 00:03:44.178
And I just happen to know this.

00:03:44.178 --> 00:03:46.460
But you might wanna do it by hand.

00:03:46.460 --> 00:03:47.543
This is 1024.

00:03:51.148 --> 00:03:53.682
So, if you're taking the cube root of 243,

00:03:53.682 --> 00:03:55.682
you're saying what to the fifth power...

00:03:55.682 --> 00:03:58.983
Something to the fifth
power is equal to 243.

00:03:58.983 --> 00:03:59.881
And, if you have a sense that

00:03:59.881 --> 00:04:02.043
it's going to be an integer solution,

00:04:02.043 --> 00:04:02.900
if you think it's going to be

00:04:02.900 --> 00:04:04.979
something like a two or a three,

00:04:04.979 --> 00:04:08.281
well, then, three is probably
going to be a good guess here.

00:04:08.281 --> 00:04:10.400
If the possible answers
are gonna be decimals,

00:04:10.400 --> 00:04:12.722
then it's going to be
a lot more complicated.

00:04:12.722 --> 00:04:13.621
But that's another way.

00:04:13.621 --> 00:04:15.007
Say, hey, maybe I'll try a three.

00:04:15.007 --> 00:04:18.757
And, if you try out
three, you would get 243.


Determining whether real world model is linear or exponential

WEBVTT
Kind: captions
Language: en

00:00:00.426 --> 00:00:02.315
- [Instructor] The table
represents the cost

00:00:02.315 --> 00:00:04.069
of buying a small piece of land

00:00:04.069 --> 00:00:07.661
in a remote village since the year 1990.

00:00:07.661 --> 00:00:10.695
Which kind of function best
models this relationship?

00:00:10.695 --> 00:00:11.628
And I'm using,

00:00:11.628 --> 00:00:13.997
this is an example from
the Khan Academy exercises,

00:00:13.997 --> 00:00:16.342
and we're really trying to pick between

00:00:16.342 --> 00:00:18.159
whether a linear model

00:00:18.159 --> 00:00:20.547
or a linear function
models this relationship

00:00:20.547 --> 00:00:21.964
or an exponential

00:00:23.128 --> 00:00:25.718
model or exponential function

00:00:25.718 --> 00:00:27.737
will model this relationship.

00:00:27.737 --> 00:00:29.195
So like always, pause this video,

00:00:29.195 --> 00:00:32.301
and see if you can figure
it out on your own.

00:00:32.301 --> 00:00:34.434
All right, so now let's
think about this together.

00:00:34.434 --> 00:00:37.129
So as the time goes by around this,

00:00:37.129 --> 00:00:38.554
the time variable right over here,

00:00:38.554 --> 00:00:41.498
we see that we keep
incrementing it by two.

00:00:41.498 --> 00:00:44.248
Go from zero to two, two to four,

00:00:45.108 --> 00:00:47.141
four to six, so on and so forth.

00:00:47.141 --> 00:00:49.159
It keeps going up by two.

00:00:49.159 --> 00:00:51.886
So if this is a linear relationship,

00:00:51.886 --> 00:00:54.930
given that our change in time is constant,

00:00:54.930 --> 00:00:58.695
our change in cost should
increase by a constant amount.

00:00:58.695 --> 00:00:59.851
Doesn't have to be this constant,

00:00:59.851 --> 00:01:01.807
but it has to be a constant amount.

00:01:01.807 --> 00:01:04.299
If we were dealing with an
exponential relationship,

00:01:04.299 --> 00:01:07.523
we would multiply by the same amount

00:01:07.523 --> 00:01:10.417
for a constant change in time.

00:01:10.417 --> 00:01:11.645
Let's see what's going on here.

00:01:11.645 --> 00:01:12.648
Let's just first look at

00:01:12.648 --> 00:01:13.915
the difference between these numbers.

00:01:13.915 --> 00:01:15.748
To go from 30 to 36.9,

00:01:16.738 --> 00:01:18.905
you would have to add 6.9.

00:01:19.927 --> 00:01:22.260
And to go from 36.9 to 44.1,

00:01:24.043 --> 00:01:25.184
what do you have to add?

00:01:25.184 --> 00:01:26.851
You have to add 7.2.

00:01:28.436 --> 00:01:30.924
And now to go from 44.1

00:01:30.924 --> 00:01:31.757
to 51.1,

00:01:33.121 --> 00:01:35.454
you would have to add seven.

00:01:36.958 --> 00:01:38.917
Now to go to 51.1

00:01:38.917 --> 00:01:39.750
to 57.9,

00:01:42.225 --> 00:01:43.808
you are adding 6.8,

00:01:45.824 --> 00:01:47.404
6.8.

00:01:47.404 --> 00:01:50.904
And then finally, going from 57.9 to 65.1,

00:01:52.231 --> 00:01:54.461
let's see, this is almost eight,

00:01:54.461 --> 00:01:56.294
7.1, this is what, 7.2

00:01:57.237 --> 00:01:59.930
we're adding, plus 7.2.

00:01:59.930 --> 00:02:00.870
So you might say, "Hey, wait,

00:02:00.870 --> 00:02:03.624
"we're not adding the exact
same amount every time."

00:02:03.624 --> 00:02:05.629
But remember, this is intended to be data

00:02:05.629 --> 00:02:10.071
that you might actually get
from a real-world situation.

00:02:10.071 --> 00:02:11.945
And the data that you get
from a real-world situation

00:02:11.945 --> 00:02:14.805
will never be exactly a linear model

00:02:14.805 --> 00:02:17.034
or exactly an exponential model.

00:02:17.034 --> 00:02:19.112
But every time we add two years,

00:02:19.112 --> 00:02:21.220
it does look like we're
getting pretty close

00:02:21.220 --> 00:02:24.169
to adding 7,000 dollars in cost.

00:02:24.169 --> 00:02:26.606
6.9 is pretty close to seven.

00:02:26.606 --> 00:02:27.869
That's pretty close to seven.

00:02:27.869 --> 00:02:28.964
That is seven.

00:02:28.964 --> 00:02:30.522
This is pretty close to seven.

00:02:30.522 --> 00:02:32.261
That's pretty close to seven.

00:02:32.261 --> 00:02:35.706
So this is looking like
a linear model to me.

00:02:35.706 --> 00:02:38.351
You could test whether
it's an exponential model.

00:02:38.351 --> 00:02:42.441
You see, well, what factor
am I multiplying each time?

00:02:42.441 --> 00:02:45.448
But that doesn't seem to be as,

00:02:45.448 --> 00:02:47.913
this doesn't seem to be
growing exponentially.

00:02:47.913 --> 00:02:49.404
It doesn't seem like we're multiplying

00:02:49.404 --> 00:02:51.247
by the same factor every time.

00:02:51.247 --> 00:02:54.685
It seems like we're multiplying
by a slightly lower factor,

00:02:54.685 --> 00:02:56.634
as we get to higher cost.

00:02:56.634 --> 00:03:00.351
So the linear model seems
to be a pretty good thing.

00:03:00.351 --> 00:03:02.575
If I see every time I
increase by two years,

00:03:02.575 --> 00:03:05.882
I'm increasing cost by
6.9 or 7.2 or seven.

00:03:05.882 --> 00:03:07.335
It's pretty close to seven.

00:03:07.335 --> 00:03:09.663
So it's not exactly the cost,

00:03:09.663 --> 00:03:12.009
but the model predicts it pretty well.

00:03:12.009 --> 00:03:15.203
If you were to plot these on a,

00:03:15.203 --> 00:03:16.674
on a coordinate plane

00:03:16.674 --> 00:03:19.226
and try to connect the dots,

00:03:19.226 --> 00:03:20.940
it would look pretty close to a line,

00:03:20.940 --> 00:03:23.350
or you could draw a line
that gets pretty close

00:03:23.350 --> 00:03:24.517
to those dots.


Examples of linear and exponential relationships

WEBVTT
Kind: captions
Language: en

00:00:00.211 --> 00:00:02.983
- [Instructor] So I have two
different xy relationships

00:00:02.983 --> 00:00:04.604
being described here.

00:00:04.604 --> 00:00:06.186
And what I would like to do in this video

00:00:06.186 --> 00:00:08.231
is figure out whether each
of these relationships,

00:00:08.231 --> 00:00:10.982
whether they are either
linear relationships,

00:00:10.982 --> 00:00:13.617
exponential relationships, or neither.

00:00:13.617 --> 00:00:14.760
And like always, pause this video

00:00:14.760 --> 00:00:17.009
and see if you can figure it out yourself.

00:00:17.009 --> 00:00:20.525
So let's look at this first
relationship right over here.

00:00:20.525 --> 00:00:23.273
And the key way to tell whether
we're dealing with a linear,

00:00:23.273 --> 00:00:25.786
or exponential, or neither relationship,

00:00:25.786 --> 00:00:28.365
is think about, okay,
for given change in x,

00:00:28.365 --> 00:00:30.429
and you see, each time here,

00:00:30.429 --> 00:00:33.091
we are increasing x by the same amount.

00:00:33.091 --> 00:00:36.345
So we're increasing x by three.

00:00:36.345 --> 00:00:39.521
So given that we're increasing
x by a constant amount,

00:00:39.521 --> 00:00:43.973
by three each time, does y
increase by a constant amount?

00:00:43.973 --> 00:00:45.150
In which case, we would be dealing

00:00:45.150 --> 00:00:46.708
with a linear relationship.

00:00:46.708 --> 00:00:50.564
Or is there a constant ratio
between successive terms

00:00:50.564 --> 00:00:52.871
when you increase x by a constant amount.

00:00:52.871 --> 00:00:53.977
In which case, we would be dealing

00:00:53.977 --> 00:00:56.172
with an exponential relationship.

00:00:56.172 --> 00:00:57.271
So let's see.

00:00:57.271 --> 00:00:59.601
Here we're going from
negative two to five.

00:00:59.601 --> 00:01:01.831
So we are adding seven.

00:01:01.831 --> 00:01:04.947
When x increases by three,
y increases by seven.

00:01:04.947 --> 00:01:09.639
When x is increasing by three,
y increases by seven again.

00:01:09.639 --> 00:01:14.433
When x increases by three,
y increases by seven again.

00:01:14.433 --> 00:01:18.636
So here, it is clearly
a linear relationship.

00:01:18.636 --> 00:01:20.549
Linear relationship.

00:01:20.549 --> 00:01:23.951
In fact, you can even, relationship,

00:01:23.951 --> 00:01:26.335
you could even plot this on a line

00:01:26.335 --> 00:01:28.430
if you assume that these
are samples on a line.

00:01:28.430 --> 00:01:30.372
You could think even about
the slope of that line.

00:01:30.372 --> 00:01:32.932
For a change in x, for
a given change in x,

00:01:32.932 --> 00:01:34.659
the change in y is always constant.

00:01:34.659 --> 00:01:36.159
When our change in x is three,

00:01:36.159 --> 00:01:38.544
our change in y is always seven.

00:01:38.544 --> 00:01:41.999
So this is clearly a linear relationship.

00:01:41.999 --> 00:01:43.713
Now let's look at this one.

00:01:43.713 --> 00:01:44.546
Let's see.

00:01:44.546 --> 00:01:48.467
Looks like our x's are
changing by one each time,

00:01:48.467 --> 00:01:49.467
so plus one.

00:01:50.517 --> 00:01:52.147
Now what are y's changing by?

00:01:52.147 --> 00:01:54.803
Here it changes by two.

00:01:54.803 --> 00:01:56.720
Then it changes by six.

00:01:58.758 --> 00:02:00.648
Alright, it's clearly not linear.

00:02:00.648 --> 00:02:02.481
Then it changes by 18.

00:02:04.275 --> 00:02:05.806
Clearly not a linear relationship.

00:02:05.806 --> 00:02:08.251
If this was linear, these
would be the same amount,

00:02:08.251 --> 00:02:11.612
same delta, same change
in y for every time,

00:02:11.612 --> 00:02:14.106
'cause we have the same change in x.

00:02:14.106 --> 00:02:16.211
So let's test to see if it's exponential.

00:02:16.211 --> 00:02:17.347
If it's an exponential,

00:02:17.347 --> 00:02:19.971
for each of these constant changes in x's,

00:02:19.971 --> 00:02:22.513
when we increase x by one every time,

00:02:22.513 --> 00:02:25.731
our ratio of successive
y's should be the same.

00:02:25.731 --> 00:02:26.564
Or another way to think about it

00:02:26.564 --> 00:02:28.746
is what are we multiplying y by?

00:02:28.746 --> 00:02:30.473
So to go from one to three,

00:02:30.473 --> 00:02:33.473
you multiply, you multiply by three.

00:02:34.483 --> 00:02:38.601
To go from three to nine,
you multiply be three.

00:02:38.601 --> 00:02:42.944
To go from nine to 27,
you multiply by three.

00:02:42.944 --> 00:02:44.504
So in a situation where every time

00:02:44.504 --> 00:02:48.143
you increase x by a fixed
amount, in this case one,

00:02:48.143 --> 00:02:49.903
and the corresponding y's

00:02:49.903 --> 00:02:53.480
get multiplied by some fixed amount,

00:02:53.480 --> 00:02:57.417
then you're dealing with an
exponential relationship.

00:02:57.417 --> 00:02:58.417
Exponential.

00:03:00.158 --> 00:03:03.575
Exponential relationship right over here.


Simplifying square-root expressions

WEBVTT
Kind: captions
Language: en

00:00:00.221 --> 00:00:01.054
- [Instructor] Let's get some practice.

00:00:01.054 --> 00:00:04.761
Simplifying radical expressions
that involve variables.

00:00:04.761 --> 00:00:06.392
So let's say I have

00:00:06.392 --> 00:00:07.973
two times

00:00:07.973 --> 00:00:10.140
the square root of seven x

00:00:11.380 --> 00:00:12.297
times three

00:00:13.633 --> 00:00:17.050
times the square root of 14 x squared.

00:00:18.486 --> 00:00:20.500
Pause the video and see
if you can simplify it.

00:00:20.500 --> 00:00:22.172
Taking any perfect squares out

00:00:22.172 --> 00:00:24.064
multiplying and taking any perfect squares

00:00:24.064 --> 00:00:26.631
out of the radical sign.

00:00:26.631 --> 00:00:28.146
Well, let's first

00:00:28.146 --> 00:00:30.799
just multiply this thing.

00:00:30.799 --> 00:00:31.632
So,

00:00:31.632 --> 00:00:33.678
we can change the order of multiplication.

00:00:33.678 --> 00:00:34.765
This is going to be the same thing

00:00:34.765 --> 00:00:36.265
as two times three

00:00:37.319 --> 00:00:40.413
times the square root of seven x

00:00:40.413 --> 00:00:42.663
times the square root of 14

00:00:43.954 --> 00:00:45.533
x squared.

00:00:45.533 --> 00:00:47.060
And so this is going to be equal to

00:00:47.060 --> 00:00:47.893
six times

00:00:48.969 --> 00:00:51.168
and then the product of two radicals,

00:00:51.168 --> 00:00:53.864
you can view that as the
square root of the product.

00:00:53.864 --> 00:00:55.947
So six times the square root of

00:00:55.947 --> 00:00:56.925
and I'll

00:00:56.925 --> 00:00:58.875
actually I'll just leave it like this.

00:00:58.875 --> 00:00:59.708
Seven

00:00:59.708 --> 00:01:00.977
times

00:01:00.977 --> 00:01:02.310
x

00:01:02.310 --> 00:01:03.143
and then

00:01:03.143 --> 00:01:04.785
let me actually factor 14.

00:01:04.785 --> 00:01:05.618
14 is two

00:01:06.579 --> 00:01:08.315
times seven

00:01:08.315 --> 00:01:09.168
times

00:01:09.168 --> 00:01:10.706
x squared.

00:01:10.706 --> 00:01:11.956
Actually let me

00:01:13.757 --> 00:01:16.757
extend my radical sign a little bit.

00:01:18.196 --> 00:01:19.654
And the reason why I
didn't multiply it out.

00:01:19.654 --> 00:01:21.523
Obviously we could've
multiplied it in our head.

00:01:21.523 --> 00:01:24.106
X times x squared is x to the third.

00:01:24.106 --> 00:01:25.564
And we could've said,

00:01:25.564 --> 00:01:26.532
seven

00:01:26.532 --> 00:01:28.698
times 14 is what

00:01:28.698 --> 00:01:29.531
98.

00:01:29.531 --> 00:01:31.443
We could've done that.

00:01:31.443 --> 00:01:32.901
But when you're trying to factor out

00:01:32.901 --> 00:01:34.334
perfect squares, it's actually

00:01:34.334 --> 00:01:37.040
easier if it's in this factored form here.

00:01:37.040 --> 00:01:38.386
Especially because,

00:01:38.386 --> 00:01:39.692
from a variable point of view

00:01:39.692 --> 00:01:42.438
you can view this as a
perfect square already.

00:01:42.438 --> 00:01:43.286
And then

00:01:43.286 --> 00:01:45.085
14's not a perfect square,

00:01:45.085 --> 00:01:46.468
seven isn't a perfect square

00:01:46.468 --> 00:01:48.551
but seven times seven is.

00:01:49.591 --> 00:01:52.192
Seven times seven is a perfect square.

00:01:52.192 --> 00:01:54.252
That is 49 of course.

00:01:54.252 --> 00:01:56.383
So let's rewrite this a little bit

00:01:56.383 --> 00:01:57.283
to see what we can do.

00:01:57.283 --> 00:01:58.972
This is going to be six

00:01:58.972 --> 00:01:59.805
times

00:02:00.725 --> 00:02:01.558
and

00:02:01.558 --> 00:02:03.372
I could write it like this.

00:02:03.372 --> 00:02:04.872
The square root of

00:02:05.746 --> 00:02:07.685
let's put all the perfect squares first

00:02:07.685 --> 00:02:10.083
so seven times seven

00:02:10.083 --> 00:02:10.916
that is

00:02:10.916 --> 00:02:12.056
49

00:02:12.056 --> 00:02:13.333
that's those two.

00:02:13.333 --> 00:02:14.166
X squared

00:02:15.069 --> 00:02:16.224
49

00:02:16.224 --> 00:02:17.542
x squared

00:02:17.542 --> 00:02:18.375
and then

00:02:18.375 --> 00:02:19.208
I could

00:02:20.247 --> 00:02:23.510
once again separate the two
radicals right over here.

00:02:23.510 --> 00:02:25.013
So whatever else is left.

00:02:25.013 --> 00:02:26.116
So

00:02:26.116 --> 00:02:28.421
I've already used the seven,
the seven, the x squared

00:02:28.421 --> 00:02:30.046
I have a two x left.

00:02:30.046 --> 00:02:31.608
Times two x.

00:02:31.608 --> 00:02:32.750
Hopefully you'll appreciate
that these two things

00:02:32.750 --> 00:02:33.583
are equivalent.

00:02:33.583 --> 00:02:35.345
I could've put one big radical sign

00:02:35.345 --> 00:02:37.493
over 49 x squared times two x

00:02:37.493 --> 00:02:39.397
which would've been exactly
what you have there,

00:02:39.397 --> 00:02:42.694
but, if you're taking the
radical of the product of things,

00:02:42.694 --> 00:02:43.750
that's the same thing as

00:02:43.750 --> 00:02:45.788
the product of the radicals.

00:02:45.788 --> 00:02:48.563
It's come straight out of
our exponent properties.

00:02:48.563 --> 00:02:50.146
But what's valuable about this is

00:02:50.146 --> 00:02:51.733
we now see this is six

00:02:51.733 --> 00:02:53.074
times

00:02:53.074 --> 00:02:53.907
now

00:02:53.907 --> 00:02:54.781
we can take the

00:02:54.781 --> 00:02:56.394
the square root of 49 x squared

00:02:56.394 --> 00:02:57.538
this is going to seven x

00:02:57.538 --> 00:03:00.423
square root of 49 is seven

00:03:00.423 --> 00:03:03.058
square root of x squared

00:03:03.058 --> 00:03:04.939
is going to be x

00:03:04.939 --> 00:03:06.257
and then we multiply that

00:03:06.257 --> 00:03:08.771
times the square root of two x

00:03:08.771 --> 00:03:11.743
times the square root of two x

00:03:11.743 --> 00:03:13.177
and so now we're in the home stretch.

00:03:13.177 --> 00:03:15.094
Six times seven is 42 x

00:03:16.770 --> 00:03:19.225
times the square root of two x

00:03:19.225 --> 00:03:20.627
and the key thing to appreciate is

00:03:20.627 --> 00:03:23.527
I keep using this property

00:03:23.527 --> 00:03:24.360
that

00:03:24.360 --> 00:03:25.710
a radical of products

00:03:25.710 --> 00:03:27.306
or the square root of products

00:03:27.306 --> 00:03:29.514
is the same thing as a
product of the square root.

00:03:29.514 --> 00:03:31.596
So even this step that I did here,

00:03:31.596 --> 00:03:34.249
if you wanted, you could've
had an intermediary step.

00:03:34.249 --> 00:03:38.974
You could've said that the
square root of 49 x squared

00:03:38.974 --> 00:03:40.443
is the same thing

00:03:40.443 --> 00:03:42.846
as square root of 49

00:03:42.846 --> 00:03:45.304
times the square root of x squared

00:03:45.304 --> 00:03:48.218
which would get us square
root of 49 is seven

00:03:48.218 --> 00:03:51.213
square root of x squared is x

00:03:51.213 --> 00:03:52.862
right over there.

00:03:52.862 --> 00:03:54.261
Let's do

00:03:54.261 --> 00:03:56.761
let's do another one of these.

00:03:58.922 --> 00:04:00.170
So let's say

00:04:00.170 --> 00:04:01.488
I have

00:04:01.488 --> 00:04:03.375
square root of two a

00:04:03.375 --> 00:04:04.208
times

00:04:04.208 --> 00:04:05.685
the square root of

00:04:05.685 --> 00:04:07.839
14 a to the third

00:04:07.839 --> 00:04:10.463
times the square root of five a.

00:04:10.463 --> 00:04:11.818
So like always, pause this video

00:04:11.818 --> 00:04:13.742
and see if you can
simplify this on your own.

00:04:13.742 --> 00:04:16.047
Multiply them and then take
all the perfect squares

00:04:16.047 --> 00:04:17.492
out of the radical.

00:04:17.492 --> 00:04:19.031
So let's multiply first.

00:04:19.031 --> 00:04:20.430
So this is gonna be the same thing

00:04:20.430 --> 00:04:22.180
as the square root of

00:04:23.638 --> 00:04:24.638
two times 14

00:04:26.426 --> 00:04:27.343
times five.

00:04:28.226 --> 00:04:29.532
So let me actually just

00:04:29.532 --> 00:04:30.365
I'm just going to

00:04:30.365 --> 00:04:31.668
two and five are prime.

00:04:31.668 --> 00:04:33.450
14 I can factor it as

00:04:33.450 --> 00:04:34.315
two times seven

00:04:34.315 --> 00:04:35.395
so this is gonna be

00:04:35.395 --> 00:04:37.752
two times, instead of
14 I'm going to write

00:04:37.752 --> 00:04:39.002
two times seven

00:04:40.138 --> 00:04:42.344
and then times five

00:04:42.344 --> 00:04:44.828
and then we have a times a to the third

00:04:44.828 --> 00:04:46.314
times a

00:04:46.314 --> 00:04:47.777
well actually let me write that as

00:04:47.777 --> 00:04:49.739
a to the fifth.

00:04:49.739 --> 00:04:51.365
We have a to the first,
times a to the third,

00:04:51.365 --> 00:04:52.331
times a to the first,

00:04:52.331 --> 00:04:54.331
and the exponents you get a to the fifth.

00:04:54.331 --> 00:04:57.466
Now, what perfect squares do we have here?

00:04:57.466 --> 00:04:59.201
Well we already see a perfect square

00:04:59.201 --> 00:05:01.843
in terms of two times two

00:05:01.843 --> 00:05:03.834
and then a to the fifth
isn't a perfect square

00:05:03.834 --> 00:05:05.807
if you think of terms of the variable a

00:05:05.807 --> 00:05:08.658
but you can view that as a
perfect square a to the fourth

00:05:08.658 --> 00:05:09.491
times a.

00:05:10.491 --> 00:05:12.916
So let's rearrange this a little bit.

00:05:12.916 --> 00:05:15.436
And so this is going to be equal to

00:05:15.436 --> 00:05:17.287
the square root

00:05:17.287 --> 00:05:18.120
of

00:05:18.120 --> 00:05:20.381
let me put my perfect squares out front.

00:05:20.381 --> 00:05:24.097
The square root of four, two times two

00:05:24.097 --> 00:05:26.163
times a to the fourth

00:05:26.163 --> 00:05:28.659
and then let me put my non perfect squares

00:05:28.659 --> 00:05:29.492
times

00:05:30.749 --> 00:05:31.887
I have a seven

00:05:31.887 --> 00:05:32.821
a five

00:05:32.821 --> 00:05:34.526
and an a that I haven't used yet

00:05:34.526 --> 00:05:36.082
so seven times five is 35

00:05:36.082 --> 00:05:36.915
so it's 35

00:05:38.105 --> 00:05:39.289
a

00:05:39.289 --> 00:05:41.030
and now just like we said before,

00:05:41.030 --> 00:05:41.863
we could

00:05:41.863 --> 00:05:42.696
let me do it

00:05:42.696 --> 00:05:43.529
we could say hey look,

00:05:43.529 --> 00:05:46.134
this is the same thing as
the square root of four

00:05:46.134 --> 00:05:48.206
times the square root of a to the fourth

00:05:48.206 --> 00:05:50.226
it's using exponent properties

00:05:50.226 --> 00:05:51.809
and then times the square root of

00:05:51.809 --> 00:05:52.642
35 a.

00:05:54.249 --> 00:05:56.886
Now principle root of
four is positive two.

00:05:56.886 --> 00:05:58.929
You can view this as
a positive square root

00:05:58.929 --> 00:06:01.130
and then square root of a to the fourth

00:06:01.130 --> 00:06:04.160
the principle root is
going to be a squared

00:06:04.160 --> 00:06:06.412
and then we're going to have that times

00:06:06.412 --> 00:06:07.625
the square root of

00:06:07.625 --> 00:06:08.458
35 a

00:06:09.820 --> 00:06:11.567
and we're done.

00:06:11.567 --> 00:06:13.181
Let's do one more example

00:06:13.181 --> 00:06:15.718
and this time, we're going
to involve two variables

00:06:15.718 --> 00:06:16.751
which as we'll see

00:06:16.751 --> 00:06:19.584
isn't that much more complicating.

00:06:21.644 --> 00:06:23.438
So let's simplify

00:06:23.438 --> 00:06:24.663
the square root

00:06:24.663 --> 00:06:25.496
of 72

00:06:26.648 --> 00:06:28.071
x to the third

00:06:28.071 --> 00:06:29.238
z to the third

00:06:30.253 --> 00:06:31.617
so the key is can we factor

00:06:31.617 --> 00:06:33.260
72 is not a perfect square

00:06:33.260 --> 00:06:36.424
but is there a perfect
square somewhere in there?

00:06:36.424 --> 00:06:38.502
And you immediately see that
if you're trying to factor it

00:06:38.502 --> 00:06:40.169
you get 36 times two

00:06:41.695 --> 00:06:44.301
and 36 of course is a perfect square.

00:06:44.301 --> 00:06:46.246
And likewise x to the
third and z to the third

00:06:46.246 --> 00:06:47.616
are not each perfect squares

00:06:47.616 --> 00:06:50.547
but they each have an x
squared and z squared in them.

00:06:50.547 --> 00:06:52.109
So let me rewrite this.

00:06:52.109 --> 00:06:53.891
This is the same thing

00:06:53.891 --> 00:06:55.865
This is the same thing as

00:06:55.865 --> 00:06:57.159
I can write

00:06:57.159 --> 00:06:59.116
let me put all my
perfect squares up front.

00:06:59.116 --> 00:07:00.944
So I have 36

00:07:00.944 --> 00:07:02.001
36

00:07:02.001 --> 00:07:03.464
I'm gonna take an x squared out

00:07:03.464 --> 00:07:05.460
x squared I'm gonna take a z squared out

00:07:05.460 --> 00:07:06.749
z squared

00:07:06.749 --> 00:07:08.224
and then we're left with

00:07:08.224 --> 00:07:09.257
is

00:07:09.257 --> 00:07:10.424
we took a 36 out

00:07:10.424 --> 00:07:13.384
so we're left with a two times two

00:07:13.384 --> 00:07:14.748
and if we took an x squared out of this

00:07:14.748 --> 00:07:16.362
we're left with just an x

00:07:16.362 --> 00:07:18.964
x to the third divided by x squared is x

00:07:18.964 --> 00:07:20.049
two x

00:07:20.049 --> 00:07:22.441
and then z to the third
divided by z squared

00:07:22.441 --> 00:07:23.274
is just z

00:07:24.205 --> 00:07:25.099
and you can verify this

00:07:25.099 --> 00:07:26.429
multiply this all out.

00:07:26.429 --> 00:07:27.990
You should be getting

00:07:27.990 --> 00:07:30.771
you should be getting
exactly what we have here.

00:07:30.771 --> 00:07:32.733
I do that little line on
the z to differentiate

00:07:32.733 --> 00:07:35.160
so it doesn't look like my twos.

00:07:35.160 --> 00:07:37.007
36 times two is 72

00:07:37.007 --> 00:07:38.853
x squared times x is x to the third.

00:07:38.853 --> 00:07:41.552
Z squared times z is z to the third.

00:07:41.552 --> 00:07:43.553
But now this is pretty
straighforward to factor

00:07:43.553 --> 00:07:44.386
because

00:07:44.386 --> 00:07:45.583
let me just

00:07:45.583 --> 00:07:47.941
I'll do more steps than
you would probably do

00:07:47.941 --> 00:07:49.749
if you were doing it on your own

00:07:49.749 --> 00:07:51.224
but that's because

00:07:51.224 --> 00:07:53.848
the whole point here is to learn.

00:07:53.848 --> 00:07:55.026
So two

00:07:55.026 --> 00:07:55.897
x

00:07:55.897 --> 00:07:57.041
z

00:07:57.041 --> 00:07:59.270
so that's just using exponent properties.

00:07:59.270 --> 00:08:01.609
And so everything here
is a perfect square.

00:08:01.609 --> 00:08:03.333
This is going to be the square root of 36

00:08:03.333 --> 00:08:04.651
times the square root of x squared

00:08:04.651 --> 00:08:07.507
times the square root of z squared

00:08:07.507 --> 00:08:10.125
which is going to be square root of 36

00:08:10.125 --> 00:08:11.153
is

00:08:11.153 --> 00:08:13.115
principle root of 36 is six

00:08:13.115 --> 00:08:15.094
principle root of x squared is x

00:08:15.094 --> 00:08:18.357
principle root of z squared is z

00:08:18.357 --> 00:08:20.098
and we're gonna multiply that

00:08:20.098 --> 00:08:20.931
times

00:08:20.931 --> 00:08:22.060
square root

00:08:22.060 --> 00:08:23.610
of two x

00:08:23.610 --> 00:08:24.754
z

00:08:24.754 --> 00:08:25.587
and

00:08:25.587 --> 00:08:27.170
we are

00:08:27.170 --> 00:08:28.003
done.