Lifecycle of massive stars

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.140
We've already talked about
the life cycle of stars

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roughly the same mass as our
sun, give or take a little bit.

00:00:06.070 --> 00:00:07.710
What I want to do
in this video is

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talk about more massive stars.

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And when I'm talking
about massive stars,

00:00:14.080 --> 00:00:17.180
I'm talking about stars that
have masses greater than 9

00:00:17.180 --> 00:00:20.960
times the sun.

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So the general idea
is exactly the same.

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You're going to start off
with this huge cloud of mainly

00:00:27.170 --> 00:00:27.910
hydrogen.

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And now, this cloud
is going to have

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to be bigger than the clouds
that condensed to form stars

00:00:34.070 --> 00:00:34.741
like our sun.

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But you're going
to start with that,

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and eventually gravity's
going to pull it together.

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And the core of it is going
to get hot and dense enough

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for hydrogen to ignite, for
hydrogen to start fusing.

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So this is hydrogen,
and it is now fusing.

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Let me write it.

00:00:50.200 --> 00:00:51.630
It is now fusing.

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Hydrogen fusion.

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Let me write it like this.

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You now have hydrogen
fusion in the middle.

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So it's ignited,
and around it, you

00:01:03.280 --> 00:01:05.820
have just the other
material of the cloud.

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So the rest of the hydrogen.

00:01:06.990 --> 00:01:09.640
And now, since it's so
heated, it's really a plasma.

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It's really kind of a soup
of electrons and nucleuses

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as opposed to well-formed atoms,
especially close to the core.

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So now you have hydrogen fusion.

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We saw this happens at
around 10 million Kelvin.

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And I want to make
it very clear.

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Since we're talking
about more massive stars,

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even at this stage,
there's going

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to be more gravitational
pressure, even at this stage,

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during the main
sequence of the star,

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because it is more massive.

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And so this is going to
burn faster and hotter.

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So this is going to be faster
and hotter than something

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the mass of our sun.

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And so even this stage
is going to happen over

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a much shorter period of
time than for a star the mass

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of our sun.

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Our sun's life is going to be
10 or 11 billion total years.

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Here, we're going to
be talking about things

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in maybe the tens of
millions of years.

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So a factor of 1,000
shorter life span.

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But anyway, let's think
about what happens.

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And so far, just the
pattern of what happens,

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it's going to happen
faster because we

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have more pressure, more
gravity, more temperature.

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But it's going to happen
in pretty much the same way

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as what we saw with a
star the mass of the sun.

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Eventually that helium--
sorry, that hydrogen

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is going to fuse into
a helium core that's

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going to have a hydrogen
shell around it.

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It's going to have a
hydrogen shell around it,

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hydrogen fusion shell around it.

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And then you have the rest
of the star around that.

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So let me label it.

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This right here is
our helium core.

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And more and more
helium is going

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to be built up as this
hydrogen in this shell fuses.

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And in a star the size of our
sun or the mass of our sun,

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this is when it starts
to become a red giant.

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Because this core is getting
denser and denser and denser

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as more and more
helium is produced.

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And as it gets denser
and denser and denser,

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there's more and more
gravitational pressure

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being put on the hydrogen,
on this hydrogen shell

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out here, where we have
fusion still happening.

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And so that's going to release
more outward energy to push out

00:03:14.440 --> 00:03:17.836
the radius of the actual star.

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So the general
process, and we're

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going to see this as the star
gets more and more massive,

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is we're going to have heavier
and heavier elements forming

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in the core.

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Those heavier and
heavier elements,

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as the star gets
denser and denser,

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will eventually ignite,
kind of supporting the core.

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But because the core itself
is getting denser and denser

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and denser, material is getting
pushed further and further out

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with more and more energy.

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Although if the star
is massive enough,

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it's not going to be
able to be pushed out

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as far as you will have
in kind of a red giant,

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with kind of a sun-like star.

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But let's just think
about how this pattern is

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going to continue.

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So eventually, that helium,
once it gets dense enough,

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it's going to ignite and it's
going to fuse into carbon.

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And you're going to have
a carbon core forming.

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So that is carbon core.

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That's a carbon core.

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Around that, you
have a helium core.

00:04:05.650 --> 00:04:07.980
And near the center
of the helium core,

00:04:07.980 --> 00:04:10.180
you have a shell
of helium fusion--

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that's helium, not hydrogen--
turning into carbon, making

00:04:13.150 --> 00:04:15.310
that carbon core
denser and hotter.

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And then around that,
you have hydrogen fusion.

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Have to be very careful.

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You have hydrogen fusion.

00:04:21.180 --> 00:04:23.925
And then around that, you
have the rest of the star.

00:04:27.172 --> 00:04:29.380
And so this process is just
going to keep continuing.

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Eventually that carbon
is going to start fusing.

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And you're going to
have heavier and heavier

00:04:33.090 --> 00:04:34.048
elements form the core.

00:04:34.048 --> 00:04:36.130
And so this is a
depiction off of Wikipedia

00:04:36.130 --> 00:04:39.440
of a fairly mature massive star.

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And you keep
forming these shells

00:04:41.760 --> 00:04:43.760
of heavier and heavier
elements, and cores

00:04:43.760 --> 00:04:46.380
of heavier and heavier
elements until eventually, you

00:04:46.380 --> 00:04:47.310
get to iron.

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And in particular, we're
talking about iron 56.

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Iron with an atomic mass of 56.

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Here on this periodic table
that 26 is its atomic number.

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It's how many protons it has.

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56, you kind of view it
as a count of the protons

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and neutrons, although
it's not exact.

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But at this point, the reason
why you stop here is that you

00:05:08.720 --> 00:05:12.290
cannot get energy
by fusing iron.

00:05:12.290 --> 00:05:15.210
Fusing iron into heavier
elements beyond iron

00:05:15.210 --> 00:05:16.720
actually requires energy.

00:05:16.720 --> 00:05:19.070
So it would actually be
an endothermic process.

00:05:19.070 --> 00:05:23.200
So to fuse iron actually
won't help support the core.

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So what I want to do in this--
So just to be very clear,

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this is how the heavy
elements actually formed.

00:05:28.630 --> 00:05:31.500
We started with
hydrogen, hydrogen fusing

00:05:31.500 --> 00:05:34.090
into helium, helium
fusing into carbon,

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and then all of these things
in various combinations--

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and I won't go into
all the details--

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are fusing heavier
and heavier elements.

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Neon, oxygen, and you
see it right over here.

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Silicon.

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And these aren't the only
elements that are forming,

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but these are kind of the main
core elements that are forming.

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But along the way, you have
all this other stuff, lithium,

00:05:50.236 --> 00:05:51.070
beryllium, boron.

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All of this other
stuff is also forming.

00:05:53.520 --> 00:05:57.330
So this is how you form
elements up to iron 56.

00:05:57.330 --> 00:06:00.980
And also, this is actually how
you can form up to nickel 56,

00:06:00.980 --> 00:06:03.600
just to be exact.

00:06:03.600 --> 00:06:05.470
There will also
be some nickel 56,

00:06:05.470 --> 00:06:07.770
which has the same
mass as iron 56,

00:06:07.770 --> 00:06:11.400
just has two fewer neutrons
and two more protons.

00:06:11.400 --> 00:06:15.590
So nickel 56 will
also form, can also

00:06:15.590 --> 00:06:17.420
be, it'll be like
a nickel-iron core.

00:06:17.420 --> 00:06:19.250
But that's about
how far a star can

00:06:19.250 --> 00:06:23.940
get, regardless of how massive
it is, at least by going

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through traditional
fusion, through

00:06:26.150 --> 00:06:28.280
the traditional
ignition mechanism.

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What I want to do
is leave you there

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just so you can think about
what might happen next,

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now that we can't fuse
this star anymore.

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And what we're actually going to
see is that it will supernova.


Sal on ABC News

WEBVTT
Kind: captions
Language: en

00:00:00.196 --> 00:00:01.363
Neal: The story of a loving dad

00:00:01.363 --> 00:00:03.236
from Northern California who happens

00:00:03.236 --> 00:00:04.734
to be pretty good at math,

00:00:04.734 --> 00:00:06.115
but who knew that when he was asked

00:00:06.115 --> 00:00:07.276
to help his nieces get through a

00:00:07.276 --> 00:00:09.442
few equations, he would end up providing

00:00:09.442 --> 00:00:10.859
the answer for tens of thousands

00:00:10.859 --> 00:00:12.477
of students everywhere.

00:00:12.477 --> 00:00:15.713
Neal Karlinsky with
our Person of the Week.

00:00:15.713 --> 00:00:18.360
Sal Kahn is a 33 year old stay-at-home dad

00:00:18.360 --> 00:00:19.559
who spends his mornings

00:00:19.559 --> 00:00:22.157
with his 18 month old, and the rest

00:00:22.157 --> 00:00:25.092
of his day crammed inside a walk-in closet

00:00:25.092 --> 00:00:27.770
running a school like no other.

00:00:27.770 --> 00:00:30.571
Sal: The exponent here is a
variable. It's not a constant.

00:00:30.571 --> 00:00:31.580
If it was just a constant,

00:00:31.580 --> 00:00:33.201
we would be able to use the power rule,

00:00:33.201 --> 00:00:34.328
but we can't do that.

00:00:34.328 --> 00:00:35.706
Neal: Why are you in a closet?

00:00:35.706 --> 00:00:37.872
Sal: This location in our house is

00:00:37.872 --> 00:00:39.956
as far away as you can get

00:00:39.956 --> 00:00:44.656
from my son while he's
screaming is the simple answer.

00:00:44.656 --> 00:00:46.292
Neal: The free videos are anything

00:00:46.292 --> 00:00:49.090
but flashy just a doodling electronic pen

00:00:49.090 --> 00:00:52.920
on a blackboard and
Khan's enthusiastic voice,

00:00:52.920 --> 00:00:54.648
but they've grown into one of the world's

00:00:54.648 --> 00:00:57.604
most popular educational websites,

00:00:57.604 --> 00:01:02.297
tutorials that are seen on
average 70,000 times a day.

00:01:02.297 --> 00:01:03.550
Sal: To start, one plus one,

00:01:03.550 --> 00:01:04.662
move at their own pace,

00:01:04.662 --> 00:01:07.001
pause and repeat videos as necessary.

00:01:07.001 --> 00:01:09.635
They can do exercises,
keep doing those exercises

00:01:09.635 --> 00:01:12.237
until they are 100% proficient.

00:01:12.237 --> 00:01:14.750
Neal: The thing is Kahn
was a hedge fund manager

00:01:14.750 --> 00:01:16.995
until a year ago and these lessons were

00:01:16.995 --> 00:01:18.754
just for his nieces.

00:01:18.754 --> 00:01:20.657
They went viral by accident growing

00:01:20.657 --> 00:01:23.049
so popular that Khan quit his job

00:01:23.049 --> 00:01:25.492
to run the Khan Academy,

00:01:25.492 --> 00:01:29.170
a collection of 1800 mostly
nine minute tutorials.

00:01:29.170 --> 00:01:31.910
To say they've caught
on, is an understatement.

00:01:31.910 --> 00:01:33.579
Bill: There's a website that I've

00:01:33.579 --> 00:01:35.074
just been using with my kids recently

00:01:35.074 --> 00:01:36.462
called Khan Academy.

00:01:36.462 --> 00:01:37.829
Neal: Bill Gates and his son are two

00:01:37.829 --> 00:01:40.134
of Khan's most loyal students addicted

00:01:40.134 --> 00:01:42.939
to his easy to understand presentations

00:01:42.939 --> 00:01:46.250
that can be repeated until
the student really gets it.

00:01:46.250 --> 00:01:47.877
Do you think he's a good teacher?

00:01:47.877 --> 00:01:49.721
Male: He's a really good teacher.

00:01:49.721 --> 00:01:50.883
Neal: After talking with students

00:01:50.883 --> 00:01:52.482
around the world on Skype,

00:01:52.482 --> 00:01:56.222
I decided to try it with
my seven year old, Max.

00:01:56.222 --> 00:01:57.467
Does this make sense?

00:01:57.467 --> 00:01:58.466
Max: Uh huh. (affirmative)

00:01:58.466 --> 00:01:59.875
Neal: Did you know that?

00:01:59.875 --> 00:02:01.732
Max: No, I didn't.

00:02:01.732 --> 00:02:04.385
Neal: What is it about the
way he talks that you like?

00:02:04.385 --> 00:02:07.492
Max: I don't know. It's
just kind of a mathy talk.

00:02:07.492 --> 00:02:08.727
Neal: It's a mathy talk?

00:02:08.727 --> 00:02:09.123
Max: Yeah.

00:02:09.123 --> 00:02:10.721
Neal: That's good, and you understand him.

00:02:10.721 --> 00:02:12.220
Max: Uh huh. (affirmative)

00:02:12.220 --> 00:02:12.963
Sal: I get a sense that it's working

00:02:12.963 --> 00:02:15.336
because it is so basic
that people really feel

00:02:15.336 --> 00:02:17.236
like it's a human talking to a human.

00:02:17.236 --> 00:02:18.377
Neal: Bill Gates favorite teacher

00:02:18.377 --> 00:02:22.876
is available to anyone who wants him free.

00:02:22.876 --> 00:02:24.577
So we choose Sal Kahn.
He's received millions

00:02:24.577 --> 00:02:26.136
of dollars in donations to keep

00:02:26.136 --> 00:02:30.136
those simple lessons coming.


Number sets 1

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.535
We are asked, what number set
does the number 8 belong to?

00:00:04.535 --> 00:00:05.910
So this is actually
a good review

00:00:05.910 --> 00:00:09.180
of the different sets of numbers
that we often talk about.

00:00:09.180 --> 00:00:11.129
So the first set
under consideration

00:00:11.130 --> 00:00:12.730
is the natural numbers.

00:00:16.740 --> 00:00:18.729
And these are essentially
the counting numbers,

00:00:18.730 --> 00:00:21.070
and you're not counting 0.

00:00:21.070 --> 00:00:23.050
So just if you were
actually to count objects,

00:00:23.050 --> 00:00:24.630
and you have at
least one of them,

00:00:24.630 --> 00:00:26.296
we're talking about
the natural numbers.

00:00:26.296 --> 00:00:30.400
So that would be 1, 2,
3, so on and so forth.

00:00:30.400 --> 00:00:32.400
So clearly, 8 is
a natural number.

00:00:32.400 --> 00:00:34.250
You can count up to 8 here.

00:00:34.250 --> 00:00:36.270
You could count 8 objects.

00:00:36.270 --> 00:00:39.960
So 8 is a member of
the natural numbers.

00:00:43.950 --> 00:00:46.360
The next one we
should consider, let's

00:00:46.360 --> 00:00:54.070
consider the whole
numbers right over here.

00:00:54.070 --> 00:00:56.880
And I should say
natural numbers.

00:00:56.880 --> 00:00:58.560
So let's consider
the whole numbers.

00:00:58.560 --> 00:00:59.934
The whole numbers
are essentially

00:00:59.934 --> 00:01:03.650
the same thing as
the natural numbers,

00:01:03.650 --> 00:01:05.860
but we're now
going to include 0.

00:01:05.860 --> 00:01:11.260
So this is 0, 1, 2,
3, so on and so forth.

00:01:11.260 --> 00:01:13.400
So clearly, 8 is one
of these as well.

00:01:13.400 --> 00:01:15.990
You could eventually
increment your way to 8,

00:01:15.990 --> 00:01:18.240
like you're just counting
all of the whole numbers.

00:01:18.240 --> 00:01:21.179
Another way to view this is
the non-negative numbers.

00:01:21.180 --> 00:01:24.530
So 8 clearly belongs
to this as well.

00:01:24.530 --> 00:01:26.100
So let's expand our
set a little bit.

00:01:26.100 --> 00:01:27.225
Let's think about integers.

00:01:31.280 --> 00:01:34.430
Now these are all the
numbers starting with,

00:01:34.430 --> 00:01:36.700
well you could keep counting
down, all the way up

00:01:36.700 --> 00:01:43.780
to negative 3, negative
2, negative 1, 0, 1, 2, 3,

00:01:43.780 --> 00:01:45.300
and you could just
keep going there.

00:01:45.300 --> 00:01:47.187
Now clearly, 8 is
one of these as well.

00:01:47.187 --> 00:01:48.520
You can just keep counting to 8.

00:01:48.520 --> 00:01:51.520
In fact, let me just
put our check box there.

00:01:51.520 --> 00:01:54.700
In general, you
have your integers

00:01:54.700 --> 00:01:58.850
that contain both the positive
and the negative numbers and 0,

00:01:58.850 --> 00:02:00.539
depending on
whether you consider

00:02:00.540 --> 00:02:02.720
that positive or
negative, or neither.

00:02:02.720 --> 00:02:05.970
So that's the integers,
right over here.

00:02:05.970 --> 00:02:08.465
And then the whole numbers
is a subset of the integers.

00:02:13.995 --> 00:02:15.370
So I'll draw it like this.

00:02:15.370 --> 00:02:18.010
The whole numbers
are right over here,

00:02:18.010 --> 00:02:20.170
that is the whole numbers.

00:02:20.170 --> 00:02:23.589
We've now excluded all
of the negative numbers.

00:02:23.590 --> 00:02:25.540
So these are all the
non-negative numbers.

00:02:25.540 --> 00:02:28.019
All the non-negative
integers, I should say.

00:02:28.020 --> 00:02:29.490
So these are the whole numbers.

00:02:29.490 --> 00:02:34.020
And then the natural numbers
are a subset of that.

00:02:34.020 --> 00:02:35.380
It's essentially everything.

00:02:35.380 --> 00:02:37.380
So the only thing that's
in whole numbers that's

00:02:37.380 --> 00:02:40.560
not in the natural numbers
is just the number 0.

00:02:40.560 --> 00:02:42.710
So this whole area
right here just

00:02:42.710 --> 00:02:44.340
corresponds to the number 0.

00:02:44.340 --> 00:02:46.270
So it really should
be a bit of a point.

00:02:46.270 --> 00:02:47.270
So let me make it clear.

00:02:51.180 --> 00:02:53.330
This circle is
the whole numbers,

00:02:53.330 --> 00:02:55.040
and then I have the
natural numbers,

00:02:55.040 --> 00:02:56.519
which is a subset of that.

00:02:56.520 --> 00:02:58.490
Obviously this isn't
drawn to scale.

00:02:58.490 --> 00:03:00.210
The natural numbers
is a subset of that.

00:03:00.210 --> 00:03:01.610
8 is a member of all of them.

00:03:01.610 --> 00:03:03.630
8 is sitting right over here.

00:03:03.630 --> 00:03:06.000
So it's a member of the
natural numbers, whole numbers,

00:03:06.000 --> 00:03:07.365
and the integers.

00:03:07.365 --> 00:03:09.160
Now let's keep expanding things.

00:03:09.160 --> 00:03:11.995
Let's talk about
rational numbers.

00:03:17.510 --> 00:03:23.239
Now these are numbers that
can be expressed in the form p

00:03:23.240 --> 00:03:27.090
over q, where both p
and q are integers.

00:03:27.090 --> 00:03:29.330
So can 8 be expressed this way?

00:03:29.330 --> 00:03:37.420
Well, you can express
8 as 8 over 1.

00:03:37.420 --> 00:03:39.459
Or actually 16 over 2.

00:03:39.460 --> 00:03:41.620
Or you could just
keep going, 32 over 4,

00:03:41.620 --> 00:03:45.140
you can express it as a
bunch of p's over q's, where

00:03:45.140 --> 00:03:47.070
both the p and q are integers.

00:03:47.070 --> 00:03:49.140
So it's definitely
a rational number.

00:03:49.140 --> 00:03:52.739
And in fact, all of these things
over here are rational numbers.

00:03:52.740 --> 00:03:53.940
So let me draw.

00:03:53.940 --> 00:03:58.940
So this is all a subset
of rational numbers.

00:03:58.940 --> 00:04:02.130
So 8 is definitely a
member of that as well.

00:04:02.130 --> 00:04:05.920
Rational numbers, so let me
put the check box over here.

00:04:05.920 --> 00:04:08.329
Now what about
irrational numbers?

00:04:08.330 --> 00:04:11.900
Irrational numbers.

00:04:11.900 --> 00:04:14.560
Well, by definition, these are
numbers that are not rational.

00:04:14.560 --> 00:04:18.050
These are numbers that cannot
be expressed in this form,

00:04:18.050 --> 00:04:19.959
where p and q are integers.

00:04:19.959 --> 00:04:22.880
So if something is rational,
it just cannot be irrational.

00:04:22.880 --> 00:04:26.110
So 8 is not a member of
the irrational numbers.

00:04:26.110 --> 00:04:28.900
The irrational numbers are
just a completely separate set

00:04:28.900 --> 00:04:30.039
over here.

00:04:30.040 --> 00:04:31.980
So I would draw it like this.

00:04:31.980 --> 00:04:33.700
This area right over
here, this would

00:04:33.700 --> 00:04:36.150
be the irrational numbers.

00:04:36.150 --> 00:04:38.349
Irrational.

00:04:38.350 --> 00:04:42.350
Rational is not a subset of
irrational, they are exclusive.

00:04:42.350 --> 00:04:44.800
You can't be in both sets.

00:04:44.800 --> 00:04:46.830
So that's irrational
right over there.

00:04:46.830 --> 00:04:52.996
And then finally let's ask, is
8 a member of the real numbers?

00:04:52.997 --> 00:04:55.080
Now the real numbers are
essentially all of these.

00:04:55.080 --> 00:04:57.810
It's combining both the
rational and the irrational.

00:04:57.810 --> 00:05:02.590
So the real numbers is all
of this right over here.

00:05:02.590 --> 00:05:06.270
And so 8 is clearly
a member of the real.

00:05:06.270 --> 00:05:08.609
It's a member of the
real, and within the real,

00:05:08.610 --> 00:05:11.310
you either can be rational or
irrational, 8 it is rational.

00:05:11.310 --> 00:05:12.070
It's an integer.

00:05:12.070 --> 00:05:13.490
It's a whole number.

00:05:13.490 --> 00:05:15.010
And it is a natural number.

00:05:15.010 --> 00:05:17.909
So it's definitely a
member of the reals.

00:05:17.909 --> 00:05:19.450
And just to give
you might be saying,

00:05:19.450 --> 00:05:21.130
hey well, what is an
irrational number then?

00:05:21.130 --> 00:05:23.260
Can't almost every number
be represented like this?

00:05:23.260 --> 00:05:25.980
Or every number you can think
of be represented like this?

00:05:25.980 --> 00:05:29.480
And an example of maybe
the most famous example

00:05:29.480 --> 00:05:31.890
of an irrational number is pi.

00:05:31.890 --> 00:05:37.599
Pi is equal to 3.14159, and
people devote their lives

00:05:37.600 --> 00:05:39.490
to memorizing the digits of pi.

00:05:39.490 --> 00:05:42.330
But what makes this irrational
is you can't represent it

00:05:42.330 --> 00:05:47.130
as a ratio, or as a
rational expression,

00:05:47.130 --> 00:05:50.890
of integers, the way you
can for rational numbers.

00:05:50.890 --> 00:05:52.650
And this right here
is non-repeating.

00:05:55.589 --> 00:05:57.130
And if it was
repeating, you actually

00:05:57.130 --> 00:05:59.310
could express it as
a ratio of integers,

00:05:59.310 --> 00:06:01.000
and we do that in other videos.

00:06:01.000 --> 00:06:04.960
It is non-repeating
and non terminating,

00:06:04.960 --> 00:06:09.664
so you never run out digits to
the right of the decimal point.

00:06:09.664 --> 00:06:11.830
So this would be an example
of an irrational number.

00:06:11.830 --> 00:06:14.190
So pi would sit here
in the irrationals.

00:06:14.190 --> 00:06:16.870
Anyway, hopefully you
found that helpful.


Applying rational equations 1

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:04.540
Ian can rake a lawn and
bag the leaves in 5 hours.

00:00:04.540 --> 00:00:08.820
Kyandre can rake the same lawn
and bag the leaves in 3 hours.

00:00:08.820 --> 00:00:11.030
Working together,
how long would it

00:00:11.030 --> 00:00:13.019
take them to rake the
lawn and bag the leaves?

00:00:13.019 --> 00:00:14.560
So let's think about
it a little bit.

00:00:14.560 --> 00:00:19.540
It says Ian can rake a lawn and
bag of the leaves in 5 hours.

00:00:19.540 --> 00:00:22.810
So for Ian, I for Ian.

00:00:22.810 --> 00:00:32.544
For 1 lawn, he can
do 1 lawn in 5 hours.

00:00:32.544 --> 00:00:34.710
We could have also written
this as 5 hours per lawn.

00:00:34.710 --> 00:00:36.150
But we'll see that
writing it this way

00:00:36.150 --> 00:00:38.025
is more useful, because
it's actually a rate.

00:00:38.025 --> 00:00:45.840
Because this is the same
thing as 1/5 lawns per hour.

00:00:45.840 --> 00:00:48.130
Or 1/5 of a lawn per hour.

00:00:48.130 --> 00:00:51.130
That's the rate at
which Ian can rake

00:00:51.130 --> 00:00:54.640
a lawn, at 1/5 of
a lawn per hour.

00:00:54.640 --> 00:00:57.162
Now let's do the same
thing for Kyandre.

00:00:57.162 --> 00:00:59.120
And this actually probably
shouldn't be plural.

00:00:59.120 --> 00:01:01.830
1/5 lawns, 1/5 of a lawn.

00:01:01.830 --> 00:01:03.750
So let me just
erase that S right

00:01:03.750 --> 00:01:07.070
there-- 1/5 of a lawn per hour.

00:01:07.070 --> 00:01:09.660
Now let's do the same
thing for Kyandre.

00:01:09.660 --> 00:01:15.800
Kyandre can rake the same lawn
and bag the leaves in 3 hours.

00:01:15.800 --> 00:01:21.930
So for Kyandre, Kyandre can,
for 1 lawn-- I'll assume it's

00:01:21.930 --> 00:01:25.090
a boy's name-- he
can do it in 3 hours.

00:01:28.690 --> 00:01:30.420
Or if we were to
write it as a rate,

00:01:30.420 --> 00:01:38.200
this is 1/3 of a lawn per hour.

00:01:38.200 --> 00:01:40.520
Now let's think about
what the combined rate is.

00:01:40.520 --> 00:01:42.620
So let's say if we
have Ian plus Kyandre.

00:01:45.750 --> 00:01:50.910
What's going to be
their combined rate?

00:01:50.910 --> 00:01:54.670
Well, they tell us that working
together, how long would

00:01:54.670 --> 00:01:56.660
it take them to rake
and bag the leaves?

00:01:56.660 --> 00:02:01.130
So let's let let's let t be
how long it will take them

00:02:01.130 --> 00:02:01.630
together.

00:02:10.190 --> 00:02:12.370
So that's how long they
would take together.

00:02:12.370 --> 00:02:14.400
And if we say t is how
long they take together,

00:02:14.400 --> 00:02:16.050
then we could say
that combined, they

00:02:16.050 --> 00:02:20.950
will do 1 lawn in
every t hours, if we're

00:02:20.950 --> 00:02:22.240
assuming t is in hours.

00:02:22.240 --> 00:02:23.620
For every t hours.

00:02:23.620 --> 00:02:25.700
Or as a rate,
their combined rate

00:02:25.700 --> 00:02:33.110
is going to be 1/t
of a lawn per hour.

00:02:35.710 --> 00:02:37.330
So that's their combined rate.

00:02:37.330 --> 00:02:41.720
So here, we have the rate of
Ian and the rate of Kyandre,

00:02:41.720 --> 00:02:42.990
and the combined rate.

00:02:42.990 --> 00:02:44.365
So the combined
rate's just going

00:02:44.365 --> 00:02:46.110
to be the sum of
each of their rates.

00:02:46.110 --> 00:02:47.790
If he can do 1/5
of a lawn per hour

00:02:47.790 --> 00:02:49.720
and he can do 1/3
of a lawn per hour,

00:02:49.720 --> 00:02:52.400
their combined rate is going
to be 1/5 of a lawn per hour

00:02:52.400 --> 00:02:53.460
plus 1/3 of an hour.

00:02:53.460 --> 00:02:55.584
Because in an hour he'll
do 1/5, and he'll do 1/3.

00:02:55.584 --> 00:02:57.500
So you'll add those two
together to figure out

00:02:57.500 --> 00:02:59.750
how much they can do in an hour.

00:02:59.750 --> 00:03:05.086
So their combined rate is
going to be 1/5 lawn per hour.

00:03:05.086 --> 00:03:06.460
And I won't write
the units here,

00:03:06.460 --> 00:03:07.751
just because it gets redundant.

00:03:07.751 --> 00:03:09.790
1/5 lawn per hour I
could write over here.

00:03:09.790 --> 00:03:12.800
Plus 1/3 lawn per
hour for Kyandre.

00:03:16.730 --> 00:03:18.640
That's going to be
their total rate, which

00:03:18.640 --> 00:03:20.780
is 1/t lawns per hour.

00:03:23.800 --> 00:03:25.300
And now we just
have to solve for t.

00:03:25.300 --> 00:03:28.830
And we'll know the total number
of hours it will take them.

00:03:28.830 --> 00:03:29.860
So let's do that.

00:03:29.860 --> 00:03:32.910
So to do that, we just
have to add 1/5 plus 1/3.

00:03:32.910 --> 00:03:35.460
Well, we have a common
denominator of 15.

00:03:35.460 --> 00:03:44.310
So this is the same
thing as 3/15 plus 5/15

00:03:44.310 --> 00:03:45.676
is going to be equal to 1/t.

00:03:48.970 --> 00:03:50.720
And then we have a
common denominator now.

00:03:50.720 --> 00:03:52.420
So 3 plus 5 is 8.

00:03:52.420 --> 00:03:54.760
So this is going to be 8/15.

00:03:54.760 --> 00:03:56.830
I'll go over here now.

00:03:56.830 --> 00:03:59.540
So now we have 3 plus 5 is 8.

00:03:59.540 --> 00:04:04.212
Over 15 is equal to 1 over t.

00:04:04.212 --> 00:04:05.670
If we want to solve
for t, we could

00:04:05.670 --> 00:04:07.780
take the reciprocal
of both sides.

00:04:07.780 --> 00:04:13.720
So if we flip the left
side, we get 15/8.

00:04:13.720 --> 00:04:17.769
And if we flip the right
side, we get t/1, or just t.

00:04:17.769 --> 00:04:19.952
So it'll take them 15/8 hours.

00:04:19.952 --> 00:04:21.410
Or if we want that
in kind of a way

00:04:21.410 --> 00:04:25.720
that we can think about it a
little bit better, 15/8-- so t

00:04:25.720 --> 00:04:30.010
is equal to 15 over 8 hours.

00:04:30.010 --> 00:04:31.990
And I should say 15
over 8 hours per lawn.

00:04:31.990 --> 00:04:35.500
This whole time here
we had lawns per hour.

00:04:35.500 --> 00:04:38.400
This was lawn per hour.

00:04:38.400 --> 00:04:40.240
And this was here as
well, lawns per hour.

00:04:40.240 --> 00:04:43.230
When we flip it, it
becomes hours per lawn.

00:04:43.230 --> 00:04:45.070
So that's exactly what we want.

00:04:45.070 --> 00:04:50.190
But 15/8 is the same
thing as 1 and 7/8 hours.

00:04:50.190 --> 00:04:53.080
And 7/8 of an hour-- we
can get our calculator out.

00:04:56.900 --> 00:05:03.010
If we have 60 minutes
in an hour, times 7/8,

00:05:03.010 --> 00:05:05.980
we get 52.5 minutes.

00:05:05.980 --> 00:05:08.630
So this is equal to our answer.

00:05:08.630 --> 00:05:20.020
Combined, it will take them 1
hour and 52.5 minutes per lawn.

00:05:20.020 --> 00:05:23.660
Or to do this lawn right over
here, the lawn in question.

00:05:23.660 --> 00:05:25.344
Hopefully, you
found that useful.


Simplifying rational expressions 1

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:09.000
Given a rectangle with length a-squared plus 6a minus 27 and a width a-squared minus 9,

00:00:09.700 --> 00:00:16.340
Write the ratio of the width of the rectangle to its length as a simplified rational expression.

00:00:16.340 --> 00:00:23.360
So we want the ratio of the width to the length

00:00:23.360 --> 00:00:27.540
to the length of the rectangle, and they give us the expressions for each of these.

00:00:27.760 --> 00:00:33.660
The width—the expression for the width of the rectangle—is a-squared minus 9. So, the width

00:00:34.200 --> 00:00:43.700
—let me do it in this pink—is a-squared minus 9, and we want the ratio of that to the length; the ratio of the width of the rectangle to its length.

00:00:44.180 --> 00:00:54.300
The length is given right over there; it is a-squared plus 6a minus 27, they want us to simplify this.

00:00:54.300 --> 00:00:59.180
And so the best way to simplify this, whether we're dealing with expressions in the numerator or denominator, or just numbers,

00:00:59.180 --> 00:01:02.120
is we want to factor them and see if they have common factors

00:01:02.120 --> 00:01:05.020
and if they do we might be able to cancel them out. So if we

00:01:05.020 --> 00:01:06.580
factor this top expression

00:01:06.800 --> 00:01:15.420
over here—which was the expression for the width—this is of the form a-squared minus b-squared, where b-squared is 9.

00:01:15.420 --> 00:01:17.340
So this is going to be the same thing as

00:01:17.400 --> 00:01:20.360
a plus the square root of 9 times

00:01:20.620 --> 00:01:22.380
a minus the square root of 9.

00:01:22.380 --> 00:01:24.620
So this is a plus 3

00:01:24.700 --> 00:01:27.020
times a minus 3

00:01:27.020 --> 00:01:28.460
and I just recognized that

00:01:28.840 --> 00:01:37.560
from just the pattern; if you ever see something a-squared minus b-squared, it's a plus b times a minus b

00:01:37.560 --> 00:01:38.840
and you can verify that for yourself;

00:01:38.840 --> 00:01:41.580
multiply this out, you'll get a-squared minus b-squared.

00:01:41.820 --> 00:01:47.060
So, the width can be factored into a plus 3 times a minus 3.

00:01:47.060 --> 00:01:49.020
Let's see if we can do something for the denominator.

00:01:50.000 --> 00:01:53.400
So here, if we want to factor this out, we have to think of two numbers

00:01:53.600 --> 00:01:58.460
that when we add them, I get positive 6, and when I take their product, I get negative 27.

00:01:58.960 --> 00:02:02.780
Let's see, if I have positive 9 and negative 3, that would work.

00:02:03.000 --> 00:02:10.840
So, this could be factored as a plus 9 and a minus 3.

00:02:11.320 --> 00:02:19.000
9 times a is 9a, a times negative 3 is negative 3a, when you add those two middle terms together, you'll get 6a, just like that

00:02:19.720 --> 00:02:25.000
and then, 9 times negative 3 is negative 27—of course the a times a is a squared.

00:02:25.780 --> 00:02:27.300
So I've factored the two expressions

00:02:27.520 --> 00:02:29.540
and let's see if we can simplify it.

00:02:29.540 --> 00:02:36.060
And before we simplify it, because when we simplify it we lose information, let's just remember what are allowable a's here,

00:02:36.200 --> 00:02:37.980
so we don't lose that information.

00:02:38.300 --> 00:02:42.960
Are there any a values here that will make this expression undefined?

00:02:43.580 --> 00:02:47.100
Well, any a value that makes the denominator zero will make this undefined.

00:02:47.260 --> 00:02:54.680
So a cannot be equal to negative 9 or 3,

00:02:55.320 --> 00:03:01.080
'cause if a was negative 9 or 3, then the denominator would be zero; this expression would be undefined.

00:03:01.080 --> 00:03:03.960
So we have to remember this, this is part of the expression;

00:03:04.320 --> 00:03:05.780
we don't want to change this domain;

00:03:05.780 --> 00:03:08.360
we don't want to allow things that weren't allowable to begin with,

00:03:08.360 --> 00:03:10.660
so let's just remember this right over here.

00:03:10.660 --> 00:03:13.860
Now with that said, now that we've made this constraint,

00:03:13.860 --> 00:03:15.300
we can simplify it more;

00:03:15.300 --> 00:03:17.760
we can say, look we have an a minus 3 in the numerator

00:03:17.760 --> 00:03:19.960
and we have an a minus 3 in the denominator

00:03:20.180 --> 00:03:26.520
and we're assuming that a is not going to be equal to 3, so it's not like we're dividing zero over zero.

00:03:26.520 --> 00:03:28.880
So a will not be equal to 3, any other number;

00:03:28.880 --> 00:03:32.140
this will be an actual number, you divide the numerator and denominator by

00:03:32.140 --> 00:03:41.840
that same value, and we are left with a plus 3 over a plus 9

00:03:41.840 --> 00:03:44.920
and the constraint here—we don't want to forget the constraints on our domain—

00:03:45.500 --> 00:03:50.300
a cannot equal negative 9 or 3.

00:03:50.300 --> 00:03:53.460
And it's important that we write this here, because over here we lost the information

00:03:53.460 --> 00:03:55.120
that a could not be equal to 3,

00:03:55.120 --> 00:04:01.940
but in order for this to really be the same thing as this thing over here, when a was equal to 3 it wasn't defined

00:04:01.940 --> 00:04:05.880
so in order for this to be the same thing, we have to constrain the domain

00:04:06.100 --> 00:04:07.780
right over there; a cannot be equal to 3.

00:04:08.240 --> 00:04:10.200
Hopefully you found that useful.


Example 3: Finding the square of a binomial with two variables

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.972
- Find the area of a
square with side (6x-5y).

00:00:05.602 --> 00:00:09.333
Let me draw our square
and all of the sides

00:00:09.333 --> 00:00:11.249
of a square are going
to have the same length,

00:00:11.249 --> 00:00:12.894
and they're telling us that the length

00:00:12.894 --> 00:00:14.799
for each of the sides which is the same

00:00:14.799 --> 00:00:17.134
for all of them is (6x-5y).

00:00:17.134 --> 00:00:19.387
So the height would be 6x-5y,

00:00:19.387 --> 00:00:22.763
and so would the width, 6x-5y,

00:00:22.763 --> 00:00:24.587
and if we wanted to find
the area of the square

00:00:24.587 --> 00:00:27.536
we just have to multiply
the width times the height.

00:00:27.536 --> 00:00:30.895
So the area for this square
is just going to be the width,

00:00:30.895 --> 00:00:35.895
which is (6x-5y) times the height,

00:00:35.957 --> 00:00:40.957
times the height which is also (6x-5y),

00:00:40.978 --> 00:00:44.291
so we just have to multiply
these two binomials.

00:00:44.291 --> 00:00:45.998
To do this, you could either do FOIL

00:00:45.998 --> 00:00:47.644
if you like memorizing things

00:00:47.644 --> 00:00:49.535
or you could just remember
this is just applying

00:00:49.535 --> 00:00:51.375
the distributive property twice.

00:00:51.375 --> 00:00:54.043
So what we could do is
distribute this entire magenta,

00:00:54.043 --> 00:00:58.486
(6x-5y), distribute it over each

00:00:58.486 --> 00:01:02.517
of these terms, in the yellow (6x-5y).

00:01:02.517 --> 00:01:04.875
If we do that, we will get

00:01:04.875 --> 00:01:09.875
this 6x times this entire (6x-5y),

00:01:11.494 --> 00:01:16.494
so (6x-5y), and then we have -5y,

00:01:17.688 --> 00:01:20.268
- 5y times once again,

00:01:20.268 --> 00:01:25.218
the entire magenta, (6x-5y).

00:01:25.218 --> 00:01:26.816
And what does this give us?

00:01:26.816 --> 00:01:30.964
So we have, we have 6x times 6x,

00:01:30.964 --> 00:01:33.767
so when I distributed just this,

00:01:33.767 --> 00:01:36.264
I'm now doing the distributive
property for the second time,

00:01:36.264 --> 00:01:41.058
6x times 6x is 36x squared,

00:01:41.058 --> 00:01:46.058
and then when I take 6x times -5y,

00:01:48.210 --> 00:01:52.499
I get 6 times -5 is -30,

00:01:52.499 --> 00:01:56.392
and then I have an x times y, -30xy.

00:01:56.392 --> 00:01:59.357
And then I want to take,

00:01:59.357 --> 00:02:01.377
I'm trying to introduce many colors here,

00:02:01.377 --> 00:02:04.300
so I have this -5y times

00:02:04.300 --> 00:02:06.729
this 6x right over here

00:02:06.729 --> 00:02:11.024
so -5 times 6 is -30.

00:02:11.024 --> 00:02:14.745
- 30 and I have a y and
an x or an x and a y,

00:02:14.745 --> 00:02:18.776
and then finally I have my
last distribution to do,

00:02:18.776 --> 00:02:20.918
let me do that maybe in white,

00:02:20.918 --> 00:02:25.918
I have -5y times another -5y,

00:02:26.995 --> 00:02:29.684
so the negative times a
negative is a positive

00:02:29.684 --> 00:02:33.197
so it is positive, 5 times 5 is 25,

00:02:33.197 --> 00:02:36.650
y times y is y squared.

00:02:36.650 --> 00:02:40.460
And then we are almost done.

00:02:40.460 --> 00:02:42.625
Right over here, we could say,

00:02:42.625 --> 00:02:45.181
we can just add these two terms

00:02:45.181 --> 00:02:46.732
in the middle right over here,

00:02:46.732 --> 00:02:51.587
- 30xy-30xy is going to be -60xy.

00:02:53.493 --> 00:02:58.490
So you get 36x squared -60xy +25y squared.

00:03:00.998 --> 00:03:05.244
Now, there is a faster way
to do this if you recognize.

00:03:05.244 --> 00:03:08.565
If you recognize that if
I'm squaring a binomial,

00:03:08.565 --> 00:03:09.899
which is essentially
what we're doing here,

00:03:09.899 --> 00:03:11.369
this is the exact same thing

00:03:11.369 --> 00:03:15.211
as 6x-5y squared.

00:03:15.211 --> 00:03:17.408
So you might recognize a pattern.

00:03:17.408 --> 00:03:20.963
If I have (a+b) squared,
this is the same thing

00:03:20.963 --> 00:03:25.187
as (a+b) times (a+b)

00:03:25.187 --> 00:03:26.755
and if you were to multiply it out

00:03:26.755 --> 00:03:28.829
this exact same way we just did it here,

00:03:28.829 --> 00:03:32.997
the pattern here is it's a
times a which is a squared,

00:03:32.997 --> 00:03:37.981
plus a times b, +ab, plus b times a,

00:03:39.118 --> 00:03:41.827
which is also +ab, we
just switched the order,

00:03:41.827 --> 00:03:44.949
plus b squared, +b
squared so this is equal

00:03:44.949 --> 00:03:49.485
to a squared +2ab +b squared.

00:03:49.485 --> 00:03:51.500
This is kind of the fast way to look,

00:03:51.500 --> 00:03:54.250
if you're squaring any binomial,
it will be a+b squared,

00:03:54.250 --> 00:03:56.559
it will be a squared +2ab + b squared.

00:03:56.559 --> 00:03:58.355
And if you knew this ahead of time,

00:03:58.355 --> 00:03:59.876
then you could have just applied that

00:03:59.876 --> 00:04:03.313
to this squaring of the
binomial right up here

00:04:03.313 --> 00:04:05.534
so let's do it that way as well.

00:04:05.534 --> 00:04:10.495
So if we 6x, (6x-5y) squared,

00:04:11.372 --> 00:04:14.080
we could just say well, this
is going to be a squared.

00:04:14.080 --> 00:04:15.551
It's going to be a squared in which

00:04:15.551 --> 00:04:20.551
in this case is 6x squared +2ab,

00:04:20.678 --> 00:04:25.678
so that's +2 times a which is (6x)

00:04:25.743 --> 00:04:28.597
times b which is (-5y),

00:04:28.597 --> 00:04:32.573
- 5y +b squared,

00:04:32.573 --> 00:04:37.521
which is +(-5y), everything is squared.

00:04:37.521 --> 00:04:39.091
And then this will simplify too,

00:04:39.091 --> 00:04:43.373
6x squared is 36x squared plus,

00:04:43.373 --> 00:04:44.931
actually it's going to be a negative here

00:04:44.931 --> 00:04:47.151
because it's going to be 2 times 6 is 12

00:04:47.151 --> 00:04:51.816
times -5 is -60, we have x and a y,

00:04:52.599 --> 00:04:55.158
x and a y, and the -5y squared

00:04:55.158 --> 00:04:58.286
is +25y squared.

00:04:58.286 --> 00:05:00.464
So hopefully you saw
multiple ways to do this,

00:05:00.464 --> 00:05:01.984
if you saw this pattern immediately,

00:05:01.984 --> 00:05:03.273
and if you knew this pattern immediately,

00:05:03.273 --> 00:05:05.711
you could just cut to the
chase and go straight here,

00:05:05.711 --> 00:05:08.088
you wouldn't have to do
distributive property twice,

00:05:08.088 --> 00:05:10.502
although, this will never be wrong.


Multiplying Polynomials1

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:04.380
Some people use the
letters FOIL as a strategy

00:00:04.380 --> 00:00:07.870
for making sure they have
multiplied all the terms.

00:00:07.870 --> 00:00:09.920
Use the FOIL
approach to multiply

00:00:09.920 --> 00:00:12.680
3x plus 2 times 5x minus 7.

00:00:12.680 --> 00:00:14.180
I'm actually not
one of those people

00:00:14.180 --> 00:00:16.910
who likes to use
the FOIL letters,

00:00:16.910 --> 00:00:18.844
because it's just a
form of memorization

00:00:18.844 --> 00:00:20.510
as opposed to realizing
that you're just

00:00:20.510 --> 00:00:22.574
using the distributive
property twice.

00:00:22.574 --> 00:00:23.990
But with that said,
let's actually

00:00:23.990 --> 00:00:26.200
use FOIL so you see
that it actually

00:00:26.200 --> 00:00:29.460
is the same thing as using the
distributive property twice.

00:00:29.460 --> 00:00:33.150
So just to remember
FOIL, it literally stands

00:00:33.150 --> 00:00:40.830
for-- so F-O-I-L. The F
in FOIL stands for First.

00:00:40.830 --> 00:00:44.570
The O stands for Outside.

00:00:44.570 --> 00:00:47.510
The I stands for Inside.

00:00:47.510 --> 00:00:49.552
And the L stands for Last.

00:00:49.552 --> 00:00:51.260
So not only do you
have to remember FOIL,

00:00:51.260 --> 00:00:54.630
you have to remember what
each of these letters

00:00:54.630 --> 00:00:55.850
actually stand for.

00:00:55.850 --> 00:00:58.930
But let's see how that applies
to multiplying these two

00:00:58.930 --> 00:00:59.990
binomials.

00:00:59.990 --> 00:01:11.280
So we have 3x plus
2 times 5x minus 7.

00:01:11.280 --> 00:01:13.295
So we're starting
with the first,

00:01:13.295 --> 00:01:14.845
we're starting with the F.

00:01:14.845 --> 00:01:18.320
We have to multiply the first
terms in each of the binomials.

00:01:18.320 --> 00:01:22.020
So that's going
to be 3x times 5x.

00:01:25.570 --> 00:01:27.760
Then we do the outside terms.

00:01:27.760 --> 00:01:32.710
So that's going to be the
3x times the negative 7.

00:01:32.710 --> 00:01:38.610
So this is going to be
plus 3x times negative 7.

00:01:38.610 --> 00:01:41.400
Then we want to do
the inside terms.

00:01:41.400 --> 00:01:43.780
Then we want to do-- I'll
do that in green-- then

00:01:43.780 --> 00:01:45.590
we want to do the inside terms.

00:01:45.590 --> 00:01:48.360
So that's the 2 and the 5x.

00:01:48.360 --> 00:01:51.770
So that's plus 2 times 5x.

00:01:51.770 --> 00:01:53.670
And then we want to
do the last terms.

00:01:53.670 --> 00:01:56.660
And the last terms are
the 2 and the negative 7,

00:01:56.660 --> 00:02:00.510
so plus 2 times negative 7.

00:02:00.510 --> 00:02:07.589
And we end up getting 3x times
5x is 15x squared, plus--

00:02:07.589 --> 00:02:09.130
or I should say
minus because there's

00:02:09.130 --> 00:02:10.505
going to be a
negative sign here.

00:02:10.505 --> 00:02:15.980
So 3x times negative
7 is negative 21x.

00:02:15.980 --> 00:02:19.963
And then we have 2
times 5x is-- let

00:02:19.963 --> 00:02:22.710
me do that in that
same shade of green.

00:02:22.710 --> 00:02:25.340
So this is plus 10x.

00:02:25.340 --> 00:02:29.050
And then 2 times negative
7 is negative 14,

00:02:29.050 --> 00:02:30.942
or now it's minus 14.

00:02:30.942 --> 00:02:32.900
And we can add these two
middle terms together.

00:02:36.280 --> 00:02:39.775
If we have negative 21x and
then we add 10x to that,

00:02:39.775 --> 00:02:43.070
that gives us negative 11x.

00:02:43.070 --> 00:02:45.350
So this right here
is negative 11x.

00:02:45.350 --> 00:02:47.680
And then these other
terms over here.

00:02:47.680 --> 00:02:53.290
We have 15x squared
minus 11x minus 14.

00:02:53.290 --> 00:02:54.614
So we just used FOIL.

00:02:54.614 --> 00:02:56.030
What I want to do
is show you this

00:02:56.030 --> 00:02:58.630
is the same thing as applying
the distributive property

00:02:58.630 --> 00:02:59.690
twice.

00:02:59.690 --> 00:03:06.560
So if we have-- I'll
rewrite the problem-- 3x

00:03:06.560 --> 00:03:11.980
plus 2 times 5x minus 7.

00:03:11.980 --> 00:03:14.520
What we could do is
distribute the 5x minus 7.

00:03:14.520 --> 00:03:16.956
We can multiply
that times the 3x.

00:03:16.956 --> 00:03:18.330
And we could also
multiply that--

00:03:18.330 --> 00:03:19.788
and let me make it
clear when I say

00:03:19.788 --> 00:03:23.230
that, I mean this whole 5x minus
7-- we can multiply it times

00:03:23.230 --> 00:03:25.290
the 3x and by the 2.

00:03:25.290 --> 00:03:28.390
We're just distributing
it over the 3x plus 2.

00:03:28.390 --> 00:03:38.578
So we get literally
3x times 5x minus 7

00:03:38.578 --> 00:03:51.410
plus 2 times 5x minus 7.

00:03:51.410 --> 00:04:00.030
And then this gives us 3x times
5x is going to be 15x squared.

00:04:00.030 --> 00:04:04.725
3x times negative 7 is
going to be negative 21x.

00:04:08.139 --> 00:04:09.680
And remember, in
this step we're just

00:04:09.680 --> 00:04:11.610
applying the distributive
property again.

00:04:11.610 --> 00:04:17.440
So then we have 2 times 5x is
going to be 10x, so plus 10x.

00:04:17.440 --> 00:04:21.630
And then we have 2 times
negative 7 is negative 14.

00:04:21.630 --> 00:04:23.530
So we actually got
the exact same result

00:04:23.530 --> 00:04:26.000
that we did when
we applied FOIL.

00:04:26.000 --> 00:04:31.474
And when you simplify it
you get this answer again.


Example 1: Modeling with polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:04.310
Write a binomial to express
the difference between the area

00:00:04.310 --> 00:00:08.170
of a rectangle with
length p and width 2r

00:00:08.170 --> 00:00:10.890
and the area of a
circle with diameter 4r.

00:00:10.890 --> 00:00:14.180
And they tell us that
p is greater than 7r.

00:00:14.180 --> 00:00:15.830
So let's first
think about the area

00:00:15.830 --> 00:00:20.530
of a rectangle with
length p and width 2r.

00:00:20.530 --> 00:00:23.640
So this is our
rectangle right here.

00:00:23.640 --> 00:00:28.390
It has a length of p and
it has a width of 2r.

00:00:28.390 --> 00:00:29.700
So what's its area?

00:00:29.700 --> 00:00:32.090
Well, it's just going to be
the length times the width.

00:00:32.090 --> 00:00:37.680
So the area here is going to be
p-- or maybe I should say 2rp.

00:00:40.630 --> 00:00:42.145
This is the length
times the width,

00:00:42.145 --> 00:00:43.395
or the width times the length.

00:00:43.395 --> 00:00:46.444
So area is equal to
2rp for the rectangle.

00:00:46.444 --> 00:00:48.860
Now, we also want to find the
difference between this area

00:00:48.860 --> 00:00:52.875
and the area of a
circle with diameter 4r.

00:00:52.875 --> 00:00:54.750
So what's the area of
the circle going to be?

00:00:54.750 --> 00:00:59.020
So let me draw our
circle over here.

00:00:59.020 --> 00:01:00.380
So our circle looks like that.

00:01:00.380 --> 00:01:04.731
Its diameter is 4r.

00:01:04.731 --> 00:01:06.480
How do we figure out
the area of a circle?

00:01:06.480 --> 00:01:12.350
Well, area is equal
to pi r squared

00:01:12.350 --> 00:01:14.300
for a circle, where
r is a radius.

00:01:14.300 --> 00:01:15.890
They gave us the diameter.

00:01:15.890 --> 00:01:17.660
The radius is half of that.

00:01:17.660 --> 00:01:21.610
So the radius here is going to
be half this distance, or 2r.

00:01:21.610 --> 00:01:27.840
So the area of our circle
is going to be pi times 2r,

00:01:27.840 --> 00:01:28.937
the whole thing squared.

00:01:28.937 --> 00:01:30.020
This is the radius, right?

00:01:30.020 --> 00:01:31.519
So we're squaring
the entire radius.

00:01:31.519 --> 00:01:35.870
So this is going to be equal
to pi times 4 times r squared.

00:01:35.870 --> 00:01:37.490
I'm just squaring
each of these terms.

00:01:37.490 --> 00:01:42.710
Or if we were to change the
order, the area of the circle

00:01:42.710 --> 00:01:47.610
is equal to 4 pi r squared.

00:01:47.610 --> 00:01:49.630
And we want to find
the difference.

00:01:49.630 --> 00:01:51.631
So to find a difference,
It's helpful--

00:01:51.631 --> 00:01:53.630
just so we don't end up
with a negative number--

00:01:53.630 --> 00:01:55.820
to figure out which of
these two is larger.

00:01:55.820 --> 00:02:01.740
So they're telling us that
p is greater than 7 r.

00:02:01.740 --> 00:02:03.830
So let's think about this.

00:02:03.830 --> 00:02:08.800
If p is greater than 7r, then
2-- let me write it this way.

00:02:08.800 --> 00:02:12.626
We know that p is
greater than 7r.

00:02:12.626 --> 00:02:15.000
So if we're going to multiply
both sides of this equation

00:02:15.000 --> 00:02:17.520
by 2rr-- and 2r
is positive, we're

00:02:17.520 --> 00:02:20.480
dealing with positive
distances, positive lengths--

00:02:20.480 --> 00:02:23.180
so if we multiply both sides
of this equation by 2r,

00:02:23.180 --> 00:02:25.530
it shouldn't change
the equation.

00:02:25.530 --> 00:02:29.630
So multiply that by 2r, and
then multiply this by 2r.

00:02:29.630 --> 00:02:33.530
And then our equation
becomes 2rp is greater

00:02:33.530 --> 00:02:40.952
than 14r squared.

00:02:40.952 --> 00:02:42.160
Now, why is this interesting?

00:02:42.160 --> 00:02:44.560
Actually, why did I even
multiply this by 2r?

00:02:44.560 --> 00:02:46.440
Well, that's so that
this becomes the same

00:02:46.440 --> 00:02:48.149
as the area of the rectangle.

00:02:48.149 --> 00:02:49.690
So this is the area
of the rectangle.

00:02:54.260 --> 00:02:56.350
And what's 14r squared?

00:02:56.350 --> 00:03:02.310
Well, 4 times pi, is going to
get us something less than 14.

00:03:02.310 --> 00:03:03.970
This is less than 14.

00:03:03.970 --> 00:03:09.366
So this is 4 pi is less than 14.

00:03:09.366 --> 00:03:18.550
14 is 4 times 3 and 2--
let me put it this way.

00:03:18.550 --> 00:03:23.330
4 times 3.5 is equal to 14.

00:03:23.330 --> 00:03:23.940
Right?

00:03:23.940 --> 00:03:27.670
So 4 times pi, which
is less than 3.5,

00:03:27.670 --> 00:03:30.470
is going to be less than 14.

00:03:30.470 --> 00:03:32.620
So we know that
this over here is

00:03:32.620 --> 00:03:35.000
larger than this
quantity over here.

00:03:35.000 --> 00:03:38.810
It's larger than 4 pi r squared.

00:03:38.810 --> 00:03:41.120
And so we know
that this rectangle

00:03:41.120 --> 00:03:43.880
has a larger area
than the circle.

00:03:43.880 --> 00:03:45.950
So we can just subtract
the circle's area

00:03:45.950 --> 00:03:48.880
from the rectangle's area
to find the difference.

00:03:48.880 --> 00:03:51.430
So the difference is going to
be the area of the rectangle,

00:03:51.430 --> 00:03:55.020
which we already
figured out is 2rp.

00:03:55.020 --> 00:03:59.100
And we're going to subtract from
that the area of the circle.

00:03:59.100 --> 00:04:06.320
The area of the circle
is 4 pi r squared.

00:04:06.320 --> 00:04:08.750
So hopefully that made sense.

00:04:08.750 --> 00:04:11.510
And one point I want to clarify.

00:04:11.510 --> 00:04:16.730
I gave the equation of the area
of a circle to be pi r squared.

00:04:16.730 --> 00:04:20.390
And then we said that the radius
is actually 2r in this case.

00:04:20.390 --> 00:04:22.102
So I substituted 2r for r.

00:04:22.102 --> 00:04:23.560
Hopefully that
doesn't confuse you.

00:04:23.560 --> 00:04:26.600
This r is the general
term for any radius.

00:04:26.600 --> 00:04:28.740
They later told us
that the actual radius

00:04:28.740 --> 00:04:30.690
is 2 times some letter r.

00:04:30.690 --> 00:04:33.320
So I substitute that
into the formula.

00:04:33.320 --> 00:04:36.680
Anyway, hopefully you
found that useful.


Simplifying a cube root

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:04.992
We're are asked to find the
cube root of negative 343.

00:00:04.992 --> 00:00:07.480
Or another way to think
about it is some number

00:00:07.480 --> 00:00:09.860
that when I multiply it
by itself three times,

00:00:09.860 --> 00:00:11.985
I'm going to get negative 343.

00:00:11.985 --> 00:00:14.325
Or another way to view
it-- this is the same thing

00:00:14.325 --> 00:00:20.634
as negative 343
to the 1/3 power.

00:00:20.634 --> 00:00:22.300
And the best way to
do this is to really

00:00:22.300 --> 00:00:23.964
just try to factor this out.

00:00:23.964 --> 00:00:25.630
So the first thing
that we could do-- so

00:00:25.630 --> 00:00:27.516
let me just factor negative 343.

00:00:27.516 --> 00:00:28.890
So the first thing
I'd like to do

00:00:28.890 --> 00:00:30.780
is just factor out
the negative 1.

00:00:30.780 --> 00:00:35.895
So this is the same thing
as negative 1 times 343.

00:00:35.895 --> 00:00:37.310
And let's think about this.

00:00:37.310 --> 00:00:38.600
Is this divisible by 2?

00:00:38.600 --> 00:00:39.100
No.

00:00:39.100 --> 00:00:40.776
Is it divisible by 3?

00:00:40.776 --> 00:00:44.650
Let's see-- the digits do not
add up to a multiple of 3.

00:00:44.650 --> 00:00:45.850
They add up to 10.

00:00:45.850 --> 00:00:47.230
So it's not divisible by 3.

00:00:47.230 --> 00:00:49.440
Not divisible by
4 since it's odd.

00:00:49.440 --> 00:00:52.620
Not divisible by 5 because it
doesn't end with a 5 or a 0.

00:00:52.620 --> 00:00:55.750
It's not divisible by 6, because
it's not divisible by 2 or 3.

00:00:55.750 --> 00:00:57.160
Is it divisible by 7?

00:00:57.160 --> 00:00:58.220
Let's check this out.

00:00:58.220 --> 00:01:05.165
So 7 goes into 343-- 7
goes into 34 four times.

00:01:05.165 --> 00:01:09.390
4 times 7 is 28.

00:01:09.390 --> 00:01:12.530
34 minus 28 is going to be 6.

00:01:12.530 --> 00:01:14.010
Bring down the 3.

00:01:14.010 --> 00:01:17.380
7 goes into 63
exactly nine times.

00:01:17.380 --> 00:01:21.570
So then we end up-- so 9
times 7 is 63, no remainder.

00:01:21.570 --> 00:01:25.050
So this is going
to be 7 times 49.

00:01:25.050 --> 00:01:29.340
And we know that 49 is the
same thing as 7 times 7.

00:01:29.340 --> 00:01:30.660
So how can we rewrite this?

00:01:30.660 --> 00:01:35.590
This is the same thing
as taking the cube

00:01:35.590 --> 00:01:43.500
root of negative 1
times 7 times 7 times

00:01:43.500 --> 00:01:47.440
7, which is the same
thing as taking the cube

00:01:47.440 --> 00:01:54.980
root of negative 1 times the
cube root of 7 times 7 times 7.

00:01:54.980 --> 00:01:58.670
Now, what's the cube
root of negative 1?

00:01:58.670 --> 00:02:02.350
Well, negative 1 times itself
three times is negative 1.

00:02:02.350 --> 00:02:04.090
So this right here
is negative 1.

00:02:04.090 --> 00:02:05.130
You could verify it.

00:02:05.130 --> 00:02:07.710
Negative 1 times negative
1 times negative 1

00:02:07.710 --> 00:02:09.440
is indeed negative 1.

00:02:09.440 --> 00:02:10.880
This becomes positive 1.

00:02:10.880 --> 00:02:13.340
Multiply by negative 1
again, you get negative 1.

00:02:13.340 --> 00:02:15.000
So this is negative 1.

00:02:15.000 --> 00:02:18.120
And then this over here,
the cube root of 7 times

00:02:18.120 --> 00:02:20.920
7 times 7-- well, that's
just going to be 7.

00:02:20.920 --> 00:02:23.110
7 multiplied by
itself three times

00:02:23.110 --> 00:02:26.650
gives us 7 times
7 times 7 or 343.

00:02:26.650 --> 00:02:28.270
So it's going to
be negative 1 times

00:02:28.270 --> 00:02:31.470
7, which is the same
thing as negative 7.

00:02:31.470 --> 00:02:34.620
So our answer is negative 7.

00:02:34.620 --> 00:02:36.770
And we're done.


Fractional exponent expressions 3

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:04.800
We're asked to simplify
9p to the 2/3, all of that

00:00:04.800 --> 00:00:08.980
over the square
root of 4p cubed.

00:00:08.980 --> 00:00:11.230
So let's just do
this step by step.

00:00:11.230 --> 00:00:13.040
The first thing we
can do is say, well,

00:00:13.040 --> 00:00:18.640
if it's 9p to the 2/3, that's
the same thing as 9 to the 2/3,

00:00:18.640 --> 00:00:23.940
9 to the 2/3 times p to the 2/3.

00:00:23.940 --> 00:00:27.020
That's what this top part
can be simplified to.

00:00:27.020 --> 00:00:29.480
Now the bottom part, we
can rewrite the bottom part

00:00:29.480 --> 00:00:31.170
right here.

00:00:31.170 --> 00:00:33.250
The square root of
4p to the third.

00:00:33.250 --> 00:00:36.660
And this is the exact
same thing as 4p

00:00:36.660 --> 00:00:40.122
to the third to
the one half power.

00:00:40.122 --> 00:00:41.080
This is the same thing.

00:00:41.080 --> 00:00:43.050
The square root
of 4p to the third

00:00:43.050 --> 00:00:46.600
is the same thing as 4p to
the third to the 1/2 power.

00:00:46.600 --> 00:00:49.110
So given that, we can do
the same thing that we

00:00:49.110 --> 00:00:50.800
did on the numerator,
we can say,

00:00:50.800 --> 00:00:53.470
well, this is the same
thing as 4 to the 1/2.

00:00:53.470 --> 00:01:00.440
4 to the 1/2 times p to
the third to the 1/2 times

00:01:00.440 --> 00:01:05.080
p to the third to the 1/2 power.

00:01:05.080 --> 00:01:07.220
Let's see if we can
simplify this even more.

00:01:07.220 --> 00:01:09.550
So this is going to
be equal to-- let's

00:01:09.550 --> 00:01:11.750
do the denominator first here.

00:01:11.750 --> 00:01:12.989
So what's 4 to the 1/2 power?

00:01:12.989 --> 00:01:14.280
What's the principal root of 4?

00:01:14.280 --> 00:01:15.710
Well, 2 squared is 4.

00:01:15.710 --> 00:01:17.110
So it's 2.

00:01:17.110 --> 00:01:20.940
The positive root is 2, and then
p to the third, and then that

00:01:20.940 --> 00:01:21.930
to the 1/2.

00:01:21.930 --> 00:01:26.240
That's the same thing as p to
3 times 1/2, or p to the 3/2.

00:01:26.240 --> 00:01:29.670
p to the 3 over 2.

00:01:29.670 --> 00:01:32.210
9 to the 2/3.

00:01:32.210 --> 00:01:34.377
See, the cube root of
9 isn't anything easy.

00:01:34.377 --> 00:01:36.460
Well, actually, you could
think about it this way.

00:01:36.460 --> 00:01:38.400
You could say it's 9 squared.

00:01:38.400 --> 00:01:39.780
So we can view nine to 2/3.

00:01:39.780 --> 00:01:40.880
Let me do it this way.

00:01:40.880 --> 00:01:42.690
This is the best way
to think about it.

00:01:42.690 --> 00:01:44.000
This is 9 to the 2/3.

00:01:44.000 --> 00:01:45.260
One could write it.

00:01:45.260 --> 00:01:48.030
I'm kind of making
it unsimplified.

00:01:48.030 --> 00:01:50.850
But it will help us calculate
what it actually means.

00:01:50.850 --> 00:01:53.050
9 to the 2/3, the
same thing as 9

00:01:53.050 --> 00:01:57.600
squared, 9 squared
to the 1/3 power,

00:01:57.600 --> 00:02:00.640
right, because 2 times
1/3 third is 2/3.

00:02:00.640 --> 00:02:04.310
And you have that, of
course, times p to the 2/3.

00:02:04.310 --> 00:02:07.240
And this is going to be
equal to 9 squared is 81.

00:02:07.240 --> 00:02:15.060
So it's 81 to the 1/3
power times p to the 2/3.

00:02:15.060 --> 00:02:17.470
Times p to the 2/3.

00:02:17.470 --> 00:02:20.810
Now instead of writing this
2 and this p to the 3/2

00:02:20.810 --> 00:02:23.920
down here, Let me just write
it as negative exponents.

00:02:23.920 --> 00:02:26.032
Or even, let's try to
just write it like this.

00:02:26.032 --> 00:02:27.990
But 2 in the denominator
is the same thing as 1

00:02:27.990 --> 00:02:30.187
to the negative 1, or 1/2.

00:02:30.187 --> 00:02:31.770
So this right here,
if I want to write

00:02:31.770 --> 00:02:34.750
numerator, same thing as 1/2.

00:02:34.750 --> 00:02:36.380
That and that are
the same thing.

00:02:36.380 --> 00:02:38.700
And p to the 3/2
in the denominator

00:02:38.700 --> 00:02:44.050
is the same thing as
p to the negative,

00:02:44.050 --> 00:02:46.450
p to the negative 3/2.

00:02:46.450 --> 00:02:48.040
So this whole
expression so far has

00:02:48.040 --> 00:02:50.750
been simplified to
this right over here.

00:02:50.750 --> 00:02:51.790
And let copy and paste.

00:02:51.790 --> 00:02:53.900
I'll work on the next line now.

00:02:53.900 --> 00:02:57.820
So let me copy it, and paste.

00:02:57.820 --> 00:03:00.310
Bring it down right over there.

00:03:00.310 --> 00:03:03.660
Let's see if we can
simplify this even more.

00:03:03.660 --> 00:03:05.050
So let's think about 81.

00:03:05.050 --> 00:03:07.930
Let's see if we can write
it as-- we're essentially,

00:03:07.930 --> 00:03:09.707
when we take
something to cubed, we

00:03:09.707 --> 00:03:11.290
take the cube root
of something, we're

00:03:11.290 --> 00:03:13.410
taking something
to the 1/3 power.

00:03:13.410 --> 00:03:14.621
Taking the cube root of it.

00:03:14.621 --> 00:03:17.120
So we have to find some numbers,
or maybe we could factor it

00:03:17.120 --> 00:03:18.844
into things that
do have cube roots,

00:03:18.844 --> 00:03:20.010
and other things that don't.

00:03:20.010 --> 00:03:22.400
So 81 is divisible by 3.

00:03:22.400 --> 00:03:30.890
It's 3 times 27, which is 3
times 9, which is 3 times 3.

00:03:30.890 --> 00:03:35.220
So 81 is 3 to the fourth power,
or 3 multiplied by itself

00:03:35.220 --> 00:03:35.970
three times.

00:03:35.970 --> 00:03:39.130
So we could rewrite
this 81 to the 1/3.

00:03:39.130 --> 00:03:45.040
we could rewrite it as 3
times 3 times 3 to the-- well,

00:03:45.040 --> 00:03:46.040
I'll write it like this.

00:03:46.040 --> 00:03:48.650
Times 3 to the 1/3.

00:03:48.650 --> 00:03:51.090
That's the same thing as 81.

00:03:51.090 --> 00:03:53.342
And then let's think
about this these p's.

00:03:53.342 --> 00:03:55.800
And I'm doing more steps than
I would do if I were doing it

00:03:55.800 --> 00:03:58.383
on a piece of paper, because I
really want y'all to understand

00:03:58.383 --> 00:03:59.930
kind of what's going on here.

00:03:59.930 --> 00:04:03.750
So we have a p to the 2/3,
times p to the negative 3/2.

00:04:03.750 --> 00:04:11.610
So that's going to be times p
to the 2/3, 2/3 minus 3/2 power.

00:04:11.610 --> 00:04:13.110
And then so we did
the p's, and then

00:04:13.110 --> 00:04:15.430
we just have the times 1/2.

00:04:15.430 --> 00:04:17.779
Times 1/2.

00:04:17.779 --> 00:04:21.800
Now this right over here,
this is the same thing.

00:04:21.800 --> 00:04:24.910
This is the same thing
as, we could just

00:04:24.910 --> 00:04:27.180
do this as 3 times
3 times 3, times 3.

00:04:27.180 --> 00:04:31.860
So it's the same thing as 3
times 3 times 3 to the 1/3,

00:04:31.860 --> 00:04:34.180
times 3 to the 1/3.

00:04:34.180 --> 00:04:35.900
We've done this multiple times.

00:04:35.900 --> 00:04:40.040
Times the 2/3 minus 3/2.

00:04:40.040 --> 00:04:41.420
Let's just work that out.

00:04:41.420 --> 00:04:43.240
Actually, three halves
is a larger number,

00:04:43.240 --> 00:04:48.590
so let's just say it's the
negative of 3/2 minus 2/3,

00:04:48.590 --> 00:04:50.940
which is equal to the
negative if we use 6

00:04:50.940 --> 00:04:52.980
as a common denominator.

00:04:52.980 --> 00:04:55.390
If we use 6 as a
common denominator,

00:04:55.390 --> 00:05:01.310
3/2 with 6 in the
denominator, is 9/6 6 minus 4

00:05:01.310 --> 00:05:04.170
over six that's what
2/3 is equal to.

00:05:04.170 --> 00:05:07.620
So 9 minus 4 is 5/6, but then
we have this negative value.

00:05:07.620 --> 00:05:10.090
So it equals negative 5/6.

00:05:10.090 --> 00:05:14.990
So times p to the
negative 5/6 power,

00:05:14.990 --> 00:05:20.200
times 1/2 times that
last 1/2 over here.

00:05:20.200 --> 00:05:24.740
Now 3 times 3 times 3, or 27 to
the 1/3, the cube root of 27,

00:05:24.740 --> 00:05:26.060
is 3.

00:05:26.060 --> 00:05:30.170
So this right here
simplifies to three.

00:05:30.170 --> 00:05:32.320
And then 3 to the 1/3,
that's the cube root of 3.

00:05:32.320 --> 00:05:35.310
So we could say, 3
and then this right

00:05:35.310 --> 00:05:37.830
here is the cube root of 3.

00:05:37.830 --> 00:05:39.190
The cube root of 3.

00:05:39.190 --> 00:05:41.520
So we get 3 times
the cube root of 3,

00:05:41.520 --> 00:05:44.486
times p to the negative 5/6.

00:05:44.486 --> 00:05:46.610
So sometimes you might want
to write the p out here

00:05:46.610 --> 00:05:47.470
in front of the radical.

00:05:47.470 --> 00:05:48.761
That tends to be a simpler way.

00:05:48.761 --> 00:05:50.340
Actually, let me do that.

00:05:50.340 --> 00:05:52.880
So I'll write the p
to the negative 5/6, p

00:05:52.880 --> 00:05:56.180
to the negative 5/6, then
I'll write the radical.

00:05:56.180 --> 00:05:59.770
So times the cube root of 3.

00:05:59.770 --> 00:06:01.450
And then all that times 1/2.

00:06:01.450 --> 00:06:04.350
or we could say all of
that multiplying by 1/2,

00:06:04.350 --> 00:06:06.570
or just divide it by two.

00:06:06.570 --> 00:06:09.940
And we've simplified it
about as far as you can go.

00:06:09.940 --> 00:06:11.500
You could view it as 3/2.

00:06:11.500 --> 00:06:17.990
You could say 3/2 is 1.5p to
the negative 5/6 times the cube

00:06:17.990 --> 00:06:18.995
root of 3.

00:06:18.995 --> 00:06:21.370
If you wanted to, you could
calculate the cube root of 3,

00:06:21.370 --> 00:06:25.120
multiply that times 1/2, give
you some decimal number times

00:06:25.120 --> 00:06:26.730
p to the negative 5/6.

00:06:26.730 --> 00:06:30.640
And that's about as simplified
as this thing I can get.


Hubble's law

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:02.860
Over many videos now,
we've been talking

00:00:02.860 --> 00:00:06.770
about how every interstellar
object is moving away

00:00:06.770 --> 00:00:07.620
from Earth.

00:00:07.620 --> 00:00:10.530
And we've also been talking
about how the further something

00:00:10.530 --> 00:00:13.104
is away from Earth,
the faster it's moving.

00:00:13.104 --> 00:00:14.520
What I want to do
in this video is

00:00:14.520 --> 00:00:16.260
to put a little bit
of numbers behind it,

00:00:16.260 --> 00:00:19.700
or even better conceptualize
what we've been talking about.

00:00:19.700 --> 00:00:22.420
So one way to think about it
is that, if at an early stage

00:00:22.420 --> 00:00:24.910
in the universe, I were
to pick some points.

00:00:24.910 --> 00:00:28.590
So that's one point, another
point, another point,

00:00:28.590 --> 00:00:30.570
another point.

00:00:30.570 --> 00:00:33.130
Let me just pick nine points
so that I have a proper grid.

00:00:33.130 --> 00:00:36.860
So this at an early
stage in the universe.

00:00:36.860 --> 00:00:39.380
If we fast forward a
few billion years--

00:00:39.380 --> 00:00:42.690
and I'm clearly not drawing it
to scale-- all of these points

00:00:42.690 --> 00:00:45.560
have all moved away
from each other.

00:00:45.560 --> 00:00:48.180
So this point is over
here-- actually, let me

00:00:48.180 --> 00:00:54.530
draw another column,
just to make it clear.

00:00:54.530 --> 00:00:57.200
So if we fast forward
a few billion years,

00:00:57.200 --> 00:00:58.920
the universe has expanded.

00:00:58.920 --> 00:01:01.040
And so everything has
moved away from everything.

00:01:01.040 --> 00:01:03.030
And let me color
code it a little bit.

00:01:03.030 --> 00:01:05.600
Let me make this point magenta.

00:01:05.600 --> 00:01:09.070
So this point, the
magenta point is now here.

00:01:09.070 --> 00:01:13.730
This green point has now moved
away from the magenta point.

00:01:13.730 --> 00:01:17.650
And now this blue point
has now moved away

00:01:17.650 --> 00:01:20.370
from the magenta point
in that direction.

00:01:20.370 --> 00:01:21.370
And we could keep going.

00:01:21.370 --> 00:01:24.710
This yellow point is
maybe over here now.

00:01:24.710 --> 00:01:26.820
I think you get
the general idea.

00:01:26.820 --> 00:01:29.850
And I'll just draw the
other yellow points.

00:01:29.850 --> 00:01:34.250
So they've all moved
away from each other.

00:01:34.250 --> 00:01:35.500
So there's no center here.

00:01:35.500 --> 00:01:39.290
Everything is just expanding
away from things next to it.

00:01:39.290 --> 00:01:42.130
And what you can see here is not
only did this thing expand away

00:01:42.130 --> 00:01:44.230
from this, but this
thing expanded away

00:01:44.230 --> 00:01:46.300
from this even further.

00:01:46.300 --> 00:01:50.500
Because it had this expansion,
plus this expansion.

00:01:50.500 --> 00:01:55.420
Or another way to
think about it is,

00:01:55.420 --> 00:01:58.280
the apparent velocity with
which something is expanding

00:01:58.280 --> 00:02:00.620
is going to be proportional
to how far it is.

00:02:00.620 --> 00:02:04.110
Because every point in between
is also expanding away.

00:02:04.110 --> 00:02:06.640
And just to review a little
bit of the visualization

00:02:06.640 --> 00:02:08.370
of this-- one way
to think of this,

00:02:08.370 --> 00:02:11.017
if you think of the universe
as an infinite flat sheet.

00:02:11.017 --> 00:02:12.600
You can imagine that
we're just taking

00:02:12.600 --> 00:02:16.680
a sheet of, I don't
know, some type of sheet

00:02:16.680 --> 00:02:19.700
of stretchy material and
just stretching it out.

00:02:19.700 --> 00:02:21.310
We're just stretching it out.

00:02:21.310 --> 00:02:24.230
That's if we kind of imagine
a more infinite universe that

00:02:24.230 --> 00:02:25.710
just goes off in
every direction.

00:02:25.710 --> 00:02:28.100
We're just stretching
that infinite sheet out.

00:02:28.100 --> 00:02:30.894
So it has no boundaries, but
we're still stretching it out.

00:02:30.894 --> 00:02:33.310
Another way to visualize it--
and this what we did earlier

00:02:33.310 --> 00:02:38.820
on-- Is you can imagine that
the universe is the three

00:02:38.820 --> 00:02:41.350
dimensional surface of a
four dimensional sphere.

00:02:41.350 --> 00:02:44.100
Or the three dimensional
surface of a hyper-sphere.

00:02:44.100 --> 00:02:45.790
So at an early stage
in the universe,

00:02:45.790 --> 00:02:47.245
the sphere looked like this.

00:02:50.660 --> 00:02:53.310
And these points here--
that magenta point

00:02:53.310 --> 00:02:54.950
is right over here.

00:02:54.950 --> 00:02:57.990
The green point is
right over there.

00:02:57.990 --> 00:03:00.570
Then we add the
blue point up here.

00:03:00.570 --> 00:03:03.400
And then let me just draw the
rest of the yellow points.

00:03:03.400 --> 00:03:05.465
And the yellow points are here.

00:03:05.465 --> 00:03:07.215
They're all on the
surface of this sphere.

00:03:09.720 --> 00:03:12.220
Obviously I'm only dealing
with two dimensions right now

00:03:12.220 --> 00:03:14.820
and it's nearly impossible,
or maybe impossible,

00:03:14.820 --> 00:03:16.980
to imagine a three
dimensional surface of a four

00:03:16.980 --> 00:03:17.990
dimensional sphere.

00:03:17.990 --> 00:03:19.340
But the analogy holds.

00:03:19.340 --> 00:03:21.030
If this is a
surface the balloon,

00:03:21.030 --> 00:03:23.250
or the surface of a
bubble, if the bubble

00:03:23.250 --> 00:03:29.160
were to expand over a few
billion years-- and once again,

00:03:29.160 --> 00:03:30.840
not drawn to scale.

00:03:30.840 --> 00:03:33.960
So now we have a
bigger bubble here.

00:03:33.960 --> 00:03:36.430
This part of the surface
is all going to expand.

00:03:36.430 --> 00:03:38.790
So once again, you
have your magenta.

00:03:38.790 --> 00:03:41.630
You have your blue dot.

00:03:41.630 --> 00:03:45.580
You have your green
dot right over here.

00:03:45.580 --> 00:03:47.860
And then let me just
draw the rest in yellow.

00:03:47.860 --> 00:03:51.750
So they will have all
expanded away from each other

00:03:51.750 --> 00:03:53.980
on the surface of this sphere.

00:03:53.980 --> 00:03:56.050
And just to make it clear
that this is a sphere,

00:03:56.050 --> 00:03:58.197
let me draw some contour lines.

00:03:58.197 --> 00:03:59.280
So this is a contour line.

00:03:59.280 --> 00:04:00.654
Just to make it
clear that we are

00:04:00.654 --> 00:04:05.500
on the surface of
an actual sphere.

00:04:05.500 --> 00:04:07.550
Now with that out of
the way, let's think

00:04:07.550 --> 00:04:11.324
about what is the apparent
velocity with which things

00:04:11.324 --> 00:04:11.990
are moving away?

00:04:11.990 --> 00:04:14.855
And remember, we're going have
to say, not only how far things

00:04:14.855 --> 00:04:17.480
are moving away, but we're going
to say how far they are moving

00:04:17.480 --> 00:04:20.120
away from-- if the
observer is us--

00:04:20.120 --> 00:04:23.400
depending on how far
they already are.

00:04:23.400 --> 00:04:26.860
So what we're going
to do-- we could say

00:04:26.860 --> 00:04:28.750
is-- let me write this down.

00:04:28.750 --> 00:04:37.860
All objects moving
away from each other.

00:04:43.330 --> 00:04:56.270
And the apparent
relative velocity

00:04:56.270 --> 00:04:59.560
is proportional to distance.

00:05:04.667 --> 00:05:06.250
And what I've just
written down here--

00:05:06.250 --> 00:05:07.624
and this is why
I wrote it down--

00:05:07.624 --> 00:05:10.060
this is a rephrasing of,
essentially, Hubble's Law.

00:05:14.440 --> 00:05:16.485
And he came up with
this by just observing

00:05:16.485 --> 00:05:18.860
that when he looks-- especially
the further out he looks,

00:05:18.860 --> 00:05:21.580
the more redshifted objects are.

00:05:21.580 --> 00:05:23.750
And not only were they
moving faster and faster away

00:05:23.750 --> 00:05:27.050
from Earth, but they seem to
be moving faster and faster

00:05:27.050 --> 00:05:28.660
away from each other.

00:05:28.660 --> 00:05:31.080
So this is just a
restating of Hubble's Law.

00:05:31.080 --> 00:05:35.260
Or another way to say
it is, from any point,

00:05:35.260 --> 00:05:37.590
let's say from the
earth, the velocity

00:05:37.590 --> 00:05:39.510
that something
appears to be moving

00:05:39.510 --> 00:05:46.220
is going to be some
constant times the distance

00:05:46.220 --> 00:05:48.630
that it is away
from the observer.

00:05:48.630 --> 00:05:50.580
In this case, we
are the observer.

00:05:50.580 --> 00:05:53.849
And we put this little
zero-- so this H here

00:05:53.849 --> 00:05:55.015
is called Hubble's Constant.

00:05:59.140 --> 00:06:01.770
And it's a very
non-constant constant.

00:06:01.770 --> 00:06:04.230
Because this constant
will change depending

00:06:04.230 --> 00:06:07.950
on where we are in the
evolution of the universe.

00:06:07.950 --> 00:06:11.880
So we put this little zero
here, this little sub 0 right

00:06:11.880 --> 00:06:16.130
over here, to show that this
is Hubble's Constant right now.

00:06:16.130 --> 00:06:17.940
And when we talk
about distance, we're

00:06:17.940 --> 00:06:20.825
talking about the proper
distance right now.

00:06:25.246 --> 00:06:26.620
And this has to
be very important

00:06:26.620 --> 00:06:28.790
because that proper
distance is constantly

00:06:28.790 --> 00:06:30.640
changing as the
universe expands.

00:06:30.640 --> 00:06:33.484
So the now will
actually change slightly

00:06:33.484 --> 00:06:35.900
from the beginning of this
video to the end of this video.

00:06:35.900 --> 00:06:40.217
But we could roughly say in kind
of our current period of time.

00:06:40.217 --> 00:06:41.800
And when we say
proper distance, we're

00:06:41.800 --> 00:06:43.810
talking about if you
actually had rulers.

00:06:43.810 --> 00:06:46.110
And if you were to just lay
them down instantaneously--

00:06:46.110 --> 00:06:48.010
obviously we can't do
something like that.

00:06:48.010 --> 00:06:50.077
But we can imagine doing
something like that.

00:06:50.077 --> 00:06:51.660
So that's what we're
talking about it.

00:06:51.660 --> 00:06:55.320
So just to give a sense of,
or do a little bit of math

00:06:55.320 --> 00:07:04.331
of how fast things are
actually moving apart-- let

00:07:04.331 --> 00:07:05.830
me actually write
it someplace where

00:07:05.830 --> 00:07:08.410
I have more space--
the current Hubble

00:07:08.410 --> 00:07:16.860
Constant is 70.6
plus or minus 3.1.

00:07:16.860 --> 00:07:19.200
So we have observed
some variation here.

00:07:19.200 --> 00:07:23.250
There is some error to
our actual measurements.

00:07:23.250 --> 00:07:29.310
Kilometers per second
per megaparsec.

00:07:33.660 --> 00:07:39.770
And remember, a parsec is
roughly 3.2, 3.3 light years.

00:07:39.770 --> 00:07:42.100
So another way to
think about it is,

00:07:42.100 --> 00:07:47.000
if this is where we are
in the universe right now.

00:07:47.000 --> 00:07:49.890
And if this object
right over here--

00:07:49.890 --> 00:07:55.680
if this distance right over here
is one megaparsec, so 1 million

00:07:55.680 --> 00:07:56.770
parsecs.

00:07:56.770 --> 00:08:03.470
Or 3.26 million light
years from Earth.

00:08:03.470 --> 00:08:06.080
So just so we have
a sense, this is

00:08:06.080 --> 00:08:11.380
roughly 3.26 million
light years from Earth.

00:08:11.380 --> 00:08:14.510
Then this object will
appear to be moving away.

00:08:14.510 --> 00:08:16.100
Although it's not
moving in space,

00:08:16.100 --> 00:08:18.600
just the space that it's in
is stretching in such a way

00:08:18.600 --> 00:08:25.960
that it looks to be moving
at, based on its redshift,

00:08:25.960 --> 00:08:31.260
70.6 kilometers per
second away from us.

00:08:31.260 --> 00:08:33.179
So this is a huge velocity.

00:08:33.179 --> 00:08:38.419
70.6 kilometers per second.

00:08:38.419 --> 00:08:42.120
So this is a pretty
fast velocity.

00:08:42.120 --> 00:08:45.160
But you have to remember,
this is over one megaparsec.

00:08:45.160 --> 00:08:49.525
The Andromeda galaxy is
not even a megaparsec away.

00:08:49.525 --> 00:08:51.960
It is about 2.5
million light years.

00:08:51.960 --> 00:08:55.314
So it's about 0.7 or
0.8 of a megaparsec.

00:08:55.314 --> 00:08:56.730
So if you look at
a point in space

00:08:56.730 --> 00:08:59.440
a little bit further than
the Andromeda galaxy,

00:08:59.440 --> 00:09:01.520
it will look to be,
right now, receding

00:09:01.520 --> 00:09:04.652
at about 70.6
kilometers per second.

00:09:04.652 --> 00:09:06.610
But what if you were to
go twice that distance?

00:09:06.610 --> 00:09:09.721
If you were to look at something
that's almost 7 light years

00:09:09.721 --> 00:09:10.220
away?

00:09:10.220 --> 00:09:12.810
2 megaparsecs away?

00:09:12.810 --> 00:09:18.600
So if you were to look
at this object over here,

00:09:18.600 --> 00:09:20.880
how fast would that be receding?

00:09:20.880 --> 00:09:22.630
Well, if you just
look at it over here,

00:09:22.630 --> 00:09:24.860
it's 2 megaparsecs away.

00:09:24.860 --> 00:09:26.190
So it's going to be twice this.

00:09:26.190 --> 00:09:28.140
You're just going to
multiply its distance-- 2

00:09:28.140 --> 00:09:30.080
megaparsecs times this.

00:09:30.080 --> 00:09:31.970
The megaparsecs cancel out.

00:09:31.970 --> 00:09:38.889
So 70.6 times 2 is-- it's
going to look to be moving,

00:09:38.889 --> 00:09:39.930
It's not moving in space.

00:09:39.930 --> 00:09:42.010
Remember, space is
just stretching.

00:09:42.010 --> 00:09:46.220
So its velocity, It's apparent
velocity, will be 70.6 times 2.

00:09:46.220 --> 00:09:51.700
So that's 141.2
kilometers per second.

00:09:51.700 --> 00:09:54.040
And one question you
say, well how did Hubble

00:09:54.040 --> 00:09:58.080
know-- you can observe the
redshift of objects moving away

00:09:58.080 --> 00:09:58.690
from us.

00:09:58.690 --> 00:10:00.606
But how did he know that
they were moving away

00:10:00.606 --> 00:10:01.390
from each other?

00:10:01.390 --> 00:10:03.640
Well, if you were look at
the redshift of this object,

00:10:03.640 --> 00:10:06.469
and say, wow, that's moving away
is 70.6 kilometers per second.

00:10:06.469 --> 00:10:08.510
And then you were to look
at the redshift of this

00:10:08.510 --> 00:10:10.176
and say, wow, that's
moving away from us

00:10:10.176 --> 00:10:13.430
at 141.2 kilometers per second.

00:10:13.430 --> 00:10:17.310
Then you also know that these
two objects are moving away

00:10:17.310 --> 00:10:22.150
from each other at 70.6
kilometers per second.

00:10:22.150 --> 00:10:24.400
And we could keep doing this
over different distances.

00:10:24.400 --> 00:10:28.870
But hopefully this gives you
a little bit bigger sense

00:10:28.870 --> 00:10:29.370
of things.

00:10:29.370 --> 00:10:31.994
And just remember, even though
I said this is a huge distance--

00:10:31.994 --> 00:10:35.420
a megaparsec is further than
it is to the Andromeda galaxy.

00:10:35.420 --> 00:10:38.720
The Andromeda galaxy is the
nearest large galaxy to us.

00:10:38.720 --> 00:10:40.220
There are some
smaller galaxies that

00:10:40.220 --> 00:10:42.290
are closer to us that
are kind of satellite

00:10:42.290 --> 00:10:43.580
galaxies around the Milky Way.

00:10:43.580 --> 00:10:48.690
But the Andromeda is the
nearest large galaxy to us.

00:10:48.690 --> 00:10:51.300
And we also know that we're
talking about hundreds

00:10:51.300 --> 00:10:57.710
of billions of galaxies in
just the observable universe.

00:10:57.710 --> 00:10:59.910
So very quickly, as
you go near the edge

00:10:59.910 --> 00:11:03.810
of the observable
universe, these velocities,

00:11:03.810 --> 00:11:06.920
the apparent distance at which
things move are moving away

00:11:06.920 --> 00:11:11.136
from us, start to become
pretty significant.


Parsec definition

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.820
You've probably
heard the word parsec

00:00:02.820 --> 00:00:07.240
before in science fiction
movies or maybe even some things

00:00:07.240 --> 00:00:08.420
dealing with astronomy.

00:00:08.420 --> 00:00:09.878
And what I want to
do in this video

00:00:09.878 --> 00:00:13.510
is really just tell you where
the word and the definition

00:00:13.510 --> 00:00:14.770
of the word really come from.

00:00:14.770 --> 00:00:16.350
And just to kind of
cut to the chase,

00:00:16.350 --> 00:00:17.790
it's just a unit of distance.

00:00:17.790 --> 00:00:23.952
It's just about
3.26 light years.

00:00:23.952 --> 00:00:25.410
But what I want to
do is just think

00:00:25.410 --> 00:00:27.390
about where did this
weird distance come from,

00:00:27.390 --> 00:00:31.010
this distance that is
roughly 3.26 light years?

00:00:31.010 --> 00:00:40.734
It comes from the distance of
something, probably a star.

00:00:40.734 --> 00:00:42.400
But let me say
"something" because there

00:00:42.400 --> 00:00:45.180
are no stars exactly
this far away from us.

00:00:45.180 --> 00:01:01.790
The distance of something that
has a parallax angle of one

00:01:01.790 --> 00:01:05.069
arc second.

00:01:05.069 --> 00:01:08.910
And the word comes from
the "par" in parallax

00:01:08.910 --> 00:01:11.980
and the "second" in arc seconds.

00:01:11.980 --> 00:01:15.660
So it's literally par-- let me
do this in a different color.

00:01:15.660 --> 00:01:18.370
It's literally parsec.

00:01:18.370 --> 00:01:22.000
You could think of it as kind
of the parallax arc second.

00:01:22.000 --> 00:01:23.720
How far would this thing be?

00:01:23.720 --> 00:01:26.084
It turns out it's
3.26 light years.

00:01:26.084 --> 00:01:27.500
So we can actually
calculate that.

00:01:27.500 --> 00:01:29.791
And that's actually what I'm
going to do in this video.

00:01:29.791 --> 00:01:32.810
So let's say there is something.

00:01:32.810 --> 00:01:34.950
So this is the sun.

00:01:34.950 --> 00:01:38.520
This is the Earth at
some point in time.

00:01:38.520 --> 00:01:40.840
This is the Earth
six months later

00:01:40.840 --> 00:01:43.300
at the opposite
end of the orbit.

00:01:43.300 --> 00:01:46.180
And we are looking
at some distance.

00:01:46.180 --> 00:01:50.310
We are looking at some
object some distance away.

00:01:50.310 --> 00:01:54.230
We know that this distance right
here is one astronomical unit.

00:01:54.230 --> 00:01:56.950
And what we want
to do is figure out

00:01:56.950 --> 00:01:59.190
the distance of this object.

00:01:59.190 --> 00:02:03.010
And all we know is that it
has parallax angle of one arc

00:02:03.010 --> 00:02:03.510
second.

00:02:03.510 --> 00:02:05.840
So let's remind ourselves
what this means.

00:02:05.840 --> 00:02:09.380
If we're looking
right at-- remember,

00:02:09.380 --> 00:02:11.160
we're looking from
above the solar system.

00:02:11.160 --> 00:02:15.190
So the Earth is rotating in
this direction in either case.

00:02:15.190 --> 00:02:17.396
And so in this
point in the year--

00:02:17.396 --> 00:02:18.520
we don't know when this is.

00:02:18.520 --> 00:02:20.470
Depends on what star that is.

00:02:20.470 --> 00:02:24.040
At this point in the
year right at sunrise,

00:02:24.040 --> 00:02:26.890
right when we first catch the
first glimpses of the sun's

00:02:26.890 --> 00:02:33.310
light, if we look
straight up, the angle

00:02:33.310 --> 00:02:37.120
between that object in the
night sky and straight up

00:02:37.120 --> 00:02:39.210
is going to be the
parallax angle.

00:02:39.210 --> 00:02:45.350
So this is going to
be one arc second.

00:02:45.350 --> 00:02:47.950
And just to make it consistent
with the last few videos

00:02:47.950 --> 00:02:49.950
we did on parallax,
let's just visualize

00:02:49.950 --> 00:02:52.720
how that would look
in the night sky.

00:02:52.720 --> 00:02:56.400
So let me draw the
night sky over here.

00:02:56.400 --> 00:02:58.670
Let me do that in purple maybe.

00:02:58.670 --> 00:03:01.520
Let me draw the
night sky over here.

00:03:01.520 --> 00:03:03.440
This is looking straight up.

00:03:03.440 --> 00:03:07.530
This is north, south,
west, and east.

00:03:07.530 --> 00:03:10.370
And so you can imagine
in this situation,

00:03:10.370 --> 00:03:12.840
the sun is just
rising on the east.

00:03:16.110 --> 00:03:17.740
Let me make it the
color of the sun.

00:03:17.740 --> 00:03:20.410
The sun is just
rising on the east.

00:03:20.410 --> 00:03:22.670
And so this will be towards
the direction of the sun.

00:03:22.670 --> 00:03:25.939
You can imagine that to some
degree, well, this is north.

00:03:25.939 --> 00:03:27.980
North is the top of the
Earth right here, kind of

00:03:27.980 --> 00:03:29.870
pointed towards us
out of the screen.

00:03:29.870 --> 00:03:31.830
South is going into the screen.

00:03:31.830 --> 00:03:33.710
Hopefully that helps
the visualization.

00:03:33.710 --> 00:03:35.085
Or another way to
think about it,

00:03:35.085 --> 00:03:36.840
the sun is rising in the east.

00:03:36.840 --> 00:03:39.800
This is going to be towards
the direction of the sun,

00:03:39.800 --> 00:03:41.920
of a certain angle
from the center.

00:03:41.920 --> 00:03:43.870
In this case, it's
on arc second.

00:03:43.870 --> 00:03:46.510
So it's going to
be right over here.

00:03:46.510 --> 00:03:51.600
So this angle right over here
is going to be one arc second.

00:03:51.600 --> 00:03:54.150
And then if we were to
see where that object is

00:03:54.150 --> 00:03:55.980
six months later,
it'll be the opposite.

00:03:55.980 --> 00:04:00.880
We're going to be looking in
the center of the universe.

00:04:00.880 --> 00:04:03.420
Or I should say, the center of
the night sky at that point,

00:04:03.420 --> 00:04:05.030
the same direction
of the universe.

00:04:05.030 --> 00:04:06.580
The universe actually
has no center.

00:04:06.580 --> 00:04:08.267
We've talked about
that many times.

00:04:08.267 --> 00:04:10.350
If we look at the same
direction in the night sky,

00:04:10.350 --> 00:04:12.840
we will be looking
six months later.

00:04:12.840 --> 00:04:15.850
And instead of it being at
dawn, it will now be at sunset.

00:04:15.850 --> 00:04:18.829
We'll be just getting the
last glimpses of the sun.

00:04:18.829 --> 00:04:22.960
And so the sun will be
setting in the west.

00:04:25.840 --> 00:04:29.280
And so this angle--
this angle right here,

00:04:29.280 --> 00:04:32.510
which is also the same
thing as a parallax angle--

00:04:32.510 --> 00:04:34.220
will also be one arc second.

00:04:39.610 --> 00:04:43.890
So let's figure out
how far this object is.

00:04:43.890 --> 00:04:45.990
What is an actual
parsec in terms

00:04:45.990 --> 00:04:49.480
of astronomical
units or light years?

00:04:49.480 --> 00:04:51.840
So if this is one
arc second, this

00:04:51.840 --> 00:04:54.230
is going to be-- and
remember one arc second is

00:04:54.230 --> 00:05:01.490
equal to 1/3600 of a degree.

00:05:01.490 --> 00:05:09.170
So this angle right over here
is going to be 90 minus 1/3600.

00:05:09.170 --> 00:05:11.260
And we just use a little
bit of trigonometry.

00:05:11.260 --> 00:05:19.100
The tangent of this angle, the
tangent of 90 minus 1/3600,

00:05:19.100 --> 00:05:23.310
is going to be this distance
in astronomical units divided

00:05:23.310 --> 00:05:23.850
by 1.

00:05:23.850 --> 00:05:25.350
Well, you divide
anything by 1, it's

00:05:25.350 --> 00:05:27.070
just going to be that distance.

00:05:27.070 --> 00:05:30.440
So that's the distance
right over there.

00:05:30.440 --> 00:05:32.560
So we get our calculator out.

00:05:32.560 --> 00:05:44.630
And we want to find the tangent
of 90 minus 1 divided by 3,600.

00:05:44.630 --> 00:05:51.230
And we will get our distance
in astronomical units, 206,264.

00:05:51.230 --> 00:05:52.390
We're going to say 265.

00:05:52.390 --> 00:05:56.080
So this distance
over here is going

00:05:56.080 --> 00:06:01.430
to be equal to
206,265-- I'm just

00:06:01.430 --> 00:06:03.670
rounding-- astronomical units.

00:06:03.670 --> 00:06:06.970
And if we want to convert
that into light years,

00:06:06.970 --> 00:06:08.690
we just divide.

00:06:08.690 --> 00:06:17.250
So there are 63,115 astronomical
units per light year.

00:06:17.250 --> 00:06:19.590
Let me actually write it down.

00:06:19.590 --> 00:06:22.660
I don't want to confuse you
with the unit cancellation.

00:06:22.660 --> 00:06:29.620
So we're dealing with
206,265 astronomical units.

00:06:29.620 --> 00:06:32.820
And we want to multiply
that times 1 light year is

00:06:32.820 --> 00:06:37.507
equal to 63,115
astronomical units.

00:06:37.507 --> 00:06:39.715
And we want this in the
numerator and the denominator

00:06:39.715 --> 00:06:41.160
to cancel out.

00:06:41.160 --> 00:06:50.170
And so if you divide 206,265,
this number up here, by 63,115,

00:06:50.170 --> 00:06:54.530
the number of astronomical
units in a light year--

00:06:54.530 --> 00:06:58.750
let me delete that right
over there-- we get 3.2.

00:06:58.750 --> 00:07:00.490
Well, the way the
math worked out here,

00:07:00.490 --> 00:07:03.830
it rounded to 3.27 light years.

00:07:03.830 --> 00:07:10.490
So this is equal to
roughly 3.27 light years.

00:07:10.490 --> 00:07:12.880
So I should just show it's
approximate right over there.

00:07:12.880 --> 00:07:14.620
But that's where the
parsec comes from.

00:07:14.620 --> 00:07:17.240
So hopefully now you just
realize it is just a distance.

00:07:17.240 --> 00:07:20.020
But even more, you actually
understand where it comes from.

00:07:20.020 --> 00:07:24.580
It's the distance that an object
needs to be from Earth in order

00:07:24.580 --> 00:07:27.571
for it to have a parallax
angle of one arc second.

00:07:27.571 --> 00:07:29.070
And that's where
the word came from.

00:07:29.070 --> 00:07:32.420
Parallax arc second.


Stellar parallax clarification

WEBVTT
Kind: captions
Language: en

00:00:00.980 --> 00:00:03.130
I got a comment on the
video where we first

00:00:03.130 --> 00:00:06.360
introduced parallax,
especially relative to stars.

00:00:06.360 --> 00:00:08.320
Essentially asking,
how do we know

00:00:08.320 --> 00:00:10.700
that this angle and this
angle is always the same?

00:00:10.700 --> 00:00:12.290
Or how do we know
that we're always

00:00:12.290 --> 00:00:14.970
looking at an isosceles
triangle, where

00:00:14.970 --> 00:00:17.430
this side is equal to this side?

00:00:17.430 --> 00:00:19.820
It worked out for this example
that I drew right here.

00:00:19.820 --> 00:00:23.710
But what if the
star was over here?

00:00:23.710 --> 00:00:26.100
What if the star was over here.

00:00:26.100 --> 00:00:27.800
Then if you just
look at it this way.

00:00:27.800 --> 00:00:32.170
If you take at this point,
the triangle is no longer,

00:00:32.170 --> 00:00:35.900
it's clearly no longer,
an isosceles triangle.

00:00:35.900 --> 00:00:38.720
It looks more like a
scalene triangle, I guess,

00:00:38.720 --> 00:00:40.370
where all of the
sides are different.

00:00:40.370 --> 00:00:42.340
And so a lot of that
trigonometry won't apply.

00:00:42.340 --> 00:00:43.964
Because we won't be
able to assume that

00:00:43.964 --> 00:00:46.140
this is a right
triangle over here.

00:00:46.140 --> 00:00:49.100
And what I want to make
clear is that that is true.

00:00:49.100 --> 00:00:50.770
You would not be able
to pick these two

00:00:50.770 --> 00:00:53.670
points during the year.

00:00:53.670 --> 00:00:56.500
These two points in our
orbit six months apart,

00:00:56.500 --> 00:00:59.210
in order to do the same math
that we did in the last video.

00:00:59.210 --> 00:01:01.110
In order to calculate
this and still

00:01:01.110 --> 00:01:03.660
have an isosceles triangle,
what you want to do

00:01:03.660 --> 00:01:06.590
is pick two different
points six months apart.

00:01:06.590 --> 00:01:08.860
So you want to do is if
this is the sun, you want

00:01:08.860 --> 00:01:11.370
to pick two different
points six months apart,

00:01:11.370 --> 00:01:13.580
where it does form an
isosceles triangle.

00:01:13.580 --> 00:01:18.220
So if this is the distance
from the sun to this other star

00:01:18.220 --> 00:01:20.410
right over here,
you want to pick

00:01:20.410 --> 00:01:25.040
a point in Earth's orbit
around the sun here.

00:01:25.040 --> 00:01:29.140
And then another point in
the orbit six months later,

00:01:29.140 --> 00:01:32.980
which would put us
right over here.

00:01:32.980 --> 00:01:37.460
And if you do that, then we
are, now all of a sudden,

00:01:37.460 --> 00:01:40.500
we are looking at
two right triangles,

00:01:40.500 --> 00:01:44.160
if we pick those
periods correctly.

00:01:44.160 --> 00:01:47.850
And the best way to think about
whether this is a perpendicular

00:01:47.850 --> 00:01:50.450
angle, is you're going to try
to find the maximum parallax

00:01:50.450 --> 00:01:53.050
from center in each
of these time periods.

00:01:53.050 --> 00:01:55.740
Here it's going to be maximally
shifted in one direction.

00:01:55.740 --> 00:01:57.690
And then when you go to
this six months later,

00:01:57.690 --> 00:02:00.106
it's going to be maximally
shifted in the other direction.

00:02:00.106 --> 00:02:02.880
So to answer that question,
the observation is right.

00:02:02.880 --> 00:02:05.710
At exactly the middle of
summer in the middle of winter,

00:02:05.710 --> 00:02:10.000
all stars will not form
an isosceles triangle

00:02:10.000 --> 00:02:12.050
with the sun and the earth.

00:02:12.050 --> 00:02:13.730
But you could pick
other points in time

00:02:13.730 --> 00:02:15.560
around the year six
months apart where

00:02:15.560 --> 00:02:19.130
any star will form an
isosceles triangle.

00:02:19.130 --> 00:02:21.860
Hopefully you
found that helpful.