Brian Lehrer Interview with Salman Khan

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Language: en

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Brian: It's Brian Lehrer Live.

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For decades futurists have been
predicting that education would be

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revolutionized by
computers and the internet.

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First we heard lots about
putting libraries online then
computers in the classroom.

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Then universities like MIT
offering free courseware,

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but nobody figured on a
young man named Salman Khan.

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His site teaches math, science, history
and more to hundreds of thousands

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worldwide and it's all free.

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Salman: The Khan Academy is most known for
what's now about 1000 videos on Youtube.

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It starts literally at 1 + 1
= 2 and it goes all the way

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up to college level calculus and physics.

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Everything is in these very short
encapsulated videos one concept at a time.

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My cousin Nadia was visiting me from
New Orleans when I was in Boston

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and she was having trouble with math.

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She was in sixth grade at the time.

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I agreed to do this
remote tutoring with her.

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So I started putting up videos
on Youtube, and before I knew it

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random people all over the world
started watching them and sending

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me letters and testimonials and I
started getting excited about it

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and I just kept going.

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Man black jacket: I think I've learned
more about math in the past three weeks

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than I had in all four years
of my high school studies.

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Man red hat: When you watch them
you just comprehend everything.

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You just get it.

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Man black shirt: I ended up passing a
whole bunch of tests from his videos.

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Salman: One thing that kind of shocked me,

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I get a lot of letters
from the Middle East.

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I get a lot of letters from
Latin America, Ethiopia, Uganda.

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It's really reached far more students
than I could have ever imagined.

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Right now there are about 1000 videos.

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I started recording them
about three years ago.

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Now I'm doing it full-time.

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I'm hoping to be able to put out
a couple of hundred videos a year.

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I see a world where literally anyone with
access to a computer and the internet

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will be able to go to the Khan Academy
and get a world class education.

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Brian: Impressive right?

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So, let's get to know the man behind
this virtual global classroom.

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Joining us live via Skype
from Mountain View California,

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Khan Academy founder, Salman Khan.

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Hey Salman. Hello from New York.

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Thanks for doing this with us.

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Salman: Hi Brian. Good to be here.

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Brian: I gather you believe you're a
better teacher on video than in person.

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Is that correct?

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Salman: That's what my cousins
first told me when I put up
the very first Youtube video.

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I viewed it as a bit of
a backhanded compliment.

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I was doing these remote tutoring
sessions, these virtual sessions with them

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and I think what they liked
about the Youtube version
of me as opposed to the live

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version of me is that they
didn't have the stress of feeling
like their cousin in Boston

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was judging them or that they had
to answer questions immediately

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or fear feeling or looking
stupid to the instructor and
that they could watch and pause

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and repeat whatever they wanted to.

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Brian: 1500 videos, 14-million views.

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What's the genesis of this Salman?

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Salman: Just like I just said, it started
with me tutoring my cousins remotely.

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They were in New Orleans, I was in Boston
and it worked out really well for them

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when I had one or two or
three that I was tutoring.

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They started doing well so I
wanted to tutor more and more.

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I had a day job.

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I was an analyst at a hedge fund at the
time and the logistics got difficult

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and I thought well it at least would
be nice, maybe I could scale myself up

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a little bit if I made some Youtube
videos and as you just mentioned,

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their initial feedback, I thought
it would be nice to have suppliment

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something they could
use to review concepts.

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It wouldn't be as good as our live
interactions, but the first feedback

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I heard from them was that
it was actually better.

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Brian: Let's see an
example of you teaching.

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This in particular is you teaching,
of all things, differential equations.

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Salman: And what's interesting here and
how this is a departure from what we've

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learned before about regular equations.

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Let me write down a regular equation
just to remind you what they look like.

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So a regular equation if we had one
variable would look something like this.

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I don't know, X squared plus the cosine
of X is equal to the square root of X.

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I just made that up.

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Here the solution is a number or
sometimes it's a set of numbers.

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Sometimes there's more than one right?

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If you have a polynomial you
could have more than one value
of X that satisfy this equation.

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Here for a differential equation,
the solution is a function.

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Our goal is to figure out what function
of X, and here I wrote F of X explicitly

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but what function of X
explicitly satisfies this
relationship or this equation.

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So let me show you what I mean by that.

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I have my differential equations book from
college so I'm going to use that as we go.

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Brian: Ah, that high school
math feeling. There it is.

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Salman: (laughing) That was
well beyond high school math.

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Brian: I guess.

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How do you approach - I mean okay you
came out of hedge funds, whatever.

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Were you an expert in
differential equations?

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Salman: An expert is kind of a loaded word
especially when you think about things

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in academic topics.

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I'm not a cutting edge researcher at the
leading edge of differential equations.

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I have not contributed to the
field of differential equations.

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But I am someone who loved math.

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I was into math in high school, college.

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I was a math major, but I teach things
outside of my credentialed areas as well.

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The way I see it is as
long as someone has -

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I think what I bring to the table is
being able to connect the intuition behind

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a lot of concepts and be able to
communicate it in a way that I wish it was

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delivered to me the first time.

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That's really what I'm trying to do.

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Brian: And you do all of these yourself?

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All 1500 videos in every subject is you.

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Salman: It's all me.

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I joke I'm the faculty
of the Khan Academy.

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Brian: Are there subjects that
are harder for you to do well?

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Salman: You know, I like challenging
myself because when I started doing this

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a lot of people said okay you've done
some algebra, can you do calculus?

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Once you do calculus can you do chemistry?

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And once you do chemistry
can you do the humanities.

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I always like pushing the
boundaries and I just naturally
like learning different topics.

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I wouldn't say harder is the word
but it requires more prep on my part

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to do something like the French
Revolution than it does to do say the

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differential equations videos.

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Brian: Do you have a business model?

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You give these away for free right?

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Salman: Yeah. The decision was
made actually before I even quit

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my job to make this a not for profit
entity, make sure it stays focused

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on its goal which is to give
education away for free.

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The way I'm paying my bills -

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I was living off savings when I
quit my job in September in 2009,

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and until recently I was
essentially just living off savings,

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but very recently we've gotten enough
donations to support a small salary for me

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and give us a bit of our own way
and hopefully we'll get more.

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Brian: Is the paid university
concept an old fashioned concept
as far as your concerned?

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Or is this just an alternative
for a very select group of people?

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Salman: Well it's
definitely an alternative.

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I don't think it's an alternative
for a select group of people.

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When you look at the user base
there's people who are gifted students

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who want to race ahead.

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There's a lot of college students
at all different skill levels.

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There are homeschoolers using it.

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There are retirees using it.

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So I wouldn't say it's selective.

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It's a fairly broad audience.

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And it's at minimum that.

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It's at minimum kind of nice to have.

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In terms of the college experience
I think there are things the
colleges are doing right.

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I think its nice to go to a campus
that's well manicured and have

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a nice experience there, but I
think this does challenge some of

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the notions of why do we have these
300-person lectures at some university.

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Actually, at all universities you
have these 300-person lectures

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when you can get on demand
instruction at your own pace,

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when you want it.

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You don't have to take notes.
It's always there for you.

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Brian: Who are your users primarily?

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I realize it's a diverse group, but
do you find certain kinds of people,

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certain groups of people are
really hitting on your stuff
for certain uses in their lives?

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Salman: Yeah, I mean the core
users are who you would expect.

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It's I would say high school and
college students, mainly in the US,

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mainly North America.

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I'd say about 80% North America,
20% the rest of the world.

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Brian: For what? As kind of an online
Sparknotes or some other reason?

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Salman: Yeah, I think they're students
who are stuck in a math class,

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either they missed some days, they
zoned out through a couple of lectures

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or they don't connect with the
professor they have and they use this

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either as a supplement or to catch
up or really as their primary source

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of understanding the concepts.

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I get a lot of letters from people
who don't use a textbook anymore

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to learn the concepts, a few people
who don't show up at lecture anymore.

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Brian: But go ahead.
That's one core group.

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Salman: And the others, you
know the homeschoolers love it.

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I didn't think about it at first,
but it's kind of an obvious group

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that would appreciate this type of
thing both for the students directly

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and for the teachers and for the parents.

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I've actually gotten -

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Some groups are using it for teacher
training, which is a compliment.

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It is actually being
used in the classroom.

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I get letters from teachers who are
one assigning the lectures for homework

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instead of reading, and some teachers
are showing it in the classroom

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and then using that as a
starting point for discussion.

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Brian: And you're obviously very
organized about how you do this.

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We're going to show another video,
a little bit of you talking about

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how some of the math and
science videos are organized.

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Salman: Sure.

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And then when you think about
the sciences, we have physics

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and the physics playlist is mainly
I call applied algebra problems.

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So, you might want to watch physics
right around after you finish algebra.

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There's only a few videos
that require calculus.

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I'll eventually make an entire
playlist on calculus-based physics.

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Most of this is just is just algebra-based
physics so you're ready to go if you

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have a good solid
understanding of algebra.

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And then the other sciences, the
chemistry it'd be nice to have algebra

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out of the way prealgebra is
probably enough for most of it.

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Prealgebra is probably enough
for most of it, but if you have
your algebra out of the way

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then you're going to have no trouble
at all with the chemistry playlist.

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Then biology does deal with a little
bit of probability and things like that.

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So, it wouldn't hurt but that's actually,

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that doesn't even require too much
mathematics or at least the mathematics

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that I've diagramed here.

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We could put biology here.

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I guess the big takeaway is that algebra
is really important for pretty much

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anything you want to do.

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There are some genetics
problems that require some
basic level and some population

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genetics problems that
have some biology in it.

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And then all the finance
and economics that also, I
think, it assumes a base level

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of algebraic understanding.

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So the things on banking.

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Brian: So, that's, as I said
very organized, and I guess

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that's the same kind of
thing that any university

00:11:01.546 --> 00:11:05.295
would do as they're figuring out
what's a prerequisite for what.

00:11:05.295 --> 00:11:06.163
Right?

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Salman: Right. That's the general idea,
and actually this video was a biproduct

00:11:11.820 --> 00:11:15.299
of there's so much content on the site now
that I got a lot of requests from people

00:11:15.299 --> 00:11:18.737
who just were kind of lost,
what do I view after what?

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And the one area where I think this
diverges from the university model -

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The university model is very -

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I mean that diagram was kind of class
based but I think what's here is a lot

00:11:27.733 --> 00:11:31.843
more granular so you can
kind of go straight to the
concept based prerequisites

00:11:31.843 --> 00:11:33.583
as opposed to an entire class.

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Brian: Are you interested in trying to
get accredited or is that not a goal?

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Salman: It's a long-term goal maybe to
have some type of direct accreditation.

00:11:41.152 --> 00:11:43.388
I mean I can imagine a world where
actually have a software piece

00:11:43.388 --> 00:11:45.657
where we give people exercises
and things like that, and you know

00:11:45.657 --> 00:11:48.538
at the end of the day, ten years in
the future it might be a lot more

00:11:48.538 --> 00:11:52.986
informative to an employer or university
as opposed to saying you have a 3.9 GPA

00:11:52.986 --> 00:11:55.198
to actually give someone the data
of all of the work that you've done.

00:11:56.321 --> 00:11:59.654
But in the short-term, maybe
the next year or two, I do see
a reality where we can tell

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people, "Look if you've understood
all of these concepts you are ready to

00:12:02.739 --> 00:12:06.815
go take the GED and you're likely to pass
it, or if you take the SAT because of the

00:12:06.815 --> 00:12:09.179
exercises you've done we have
this level of confidence that

00:12:09.179 --> 00:12:11.621
you will get a X on the SAT."

00:12:11.653 --> 00:12:15.156
So we can view the Khan Academy
is kind of your coach and then you

00:12:15.156 --> 00:12:17.456
get your external credential
from some other source.

00:12:18.067 --> 00:12:21.225
Brian: I didn't see you in any
of the videos that we screened.

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Do your students ever see you at a smart
board or with an overhead projector

00:12:27.235 --> 00:12:28.976
or anything like that in these videos?

00:12:29.403 --> 00:12:33.317
Salman: No, and you know the form
factor started off out of me just

00:12:33.317 --> 00:12:39.176
not having a video camera, but
then I got a lot of feedback

00:12:39.176 --> 00:12:41.735
that this form factor actually works
for a lot of people even though now

00:12:41.735 --> 00:12:43.946
I probably could get a video
camera and go at a whiteboard,

00:12:43.946 --> 00:12:45.789
but people say that it
feels a lot more intimate.

00:12:45.789 --> 00:12:48.990
The instructor's face isn't a distraction.

00:12:48.990 --> 00:12:52.024
It's kind of like they
feel like I'm the voice.

00:12:52.024 --> 00:12:54.651
I'm either sitting next to them
and tutoring them or they feel like

00:12:54.651 --> 00:12:57.149
I'm actually inside of
their brain talking to them.

00:12:57.149 --> 00:12:59.402
So I think I'll just stick with that.

00:12:59.403 --> 00:13:01.445
It's actually funny because I've
gotten letters from people who imagine

00:13:01.445 --> 00:13:05.065
me in completely different ways than
I might be and they kind of project

00:13:05.065 --> 00:13:09.200
their own impressions onto
me which is kind of a neat
artifact of this form factor.

00:13:09.483 --> 00:13:13.901
Brian: You said that one of your goals
is to teach the way you always wanted

00:13:13.901 --> 00:13:15.117
to be taught.

00:13:15.360 --> 00:13:16.701
Can you put that into words?

00:13:17.423 --> 00:13:18.925
Salman: Yeah, you know I think it's -

00:13:18.925 --> 00:13:23.187
I'm finding this out more and more
the more concepts I do that there's

00:13:23.187 --> 00:13:27.961
a lot of intuitive connections between
a lot of subjects but I think the

00:13:27.961 --> 00:13:31.939
1000 page text books and then you
know to some degree the beaurocracies

00:13:31.939 --> 00:13:33.945
of the standards making organizations.

00:13:33.945 --> 00:13:38.474
They kind of chop them up to a level
that you lose all of the intuition.

00:13:38.474 --> 00:13:42.283
You lose all of the cross
subject interactions and for me,

00:13:42.283 --> 00:13:46.240
what made math fun, what made
science fun was to really explore

00:13:46.240 --> 00:13:49.189
things and see how different
concepts were connected and what

00:13:49.189 --> 00:13:51.734
a formula is actually
saying about the universe.

00:13:51.734 --> 00:13:55.483
I think people who watch the Khan
Academy they really find I'm not just

00:13:55.483 --> 00:13:56.788
throwing formulas at you.

00:13:56.788 --> 00:13:59.028
I give you an intuition for what the
formula is saying, and I also tell you

00:13:59.028 --> 00:14:02.697
what it's saying about the universe
generally, what are its implications,

00:14:02.697 --> 00:14:05.737
and also how did we get that formula.

00:14:05.737 --> 00:14:09.656
I don't just say memorize it and apply
it, and I think when you start learning

00:14:09.656 --> 00:14:12.837
that way, one it's a lot more fun
because we all actually do want

00:14:12.837 --> 00:14:16.402
to learn about the universe and it
actually builds a really, really strong

00:14:16.402 --> 00:14:19.346
foundation when you learn
more advanced topics.

00:14:19.346 --> 00:14:20.948
Brian: Which are your most popular videos?

00:14:21.692 --> 00:14:23.925
Salman: Introduction to trigonometry is -

00:14:23.925 --> 00:14:24.854
Brian: Really?

00:14:24.854 --> 00:14:28.957
Saman: That or probably
introduction to calculus.

00:14:28.957 --> 00:14:31.933
Those are probably the
two most popular videos.

00:14:31.933 --> 00:14:32.834
Brian: Why?

00:14:33.567 --> 00:14:37.231
Salman: Well, I think both people
take those courses, both people are at

00:14:37.231 --> 00:14:39.696
the same time kind of
intimidated by those courses.

00:14:39.696 --> 00:14:43.305
I think when you watch
those first two videos -

00:14:43.305 --> 00:14:45.949
The introduction to calculus is an
interesting one because I've had

00:14:45.949 --> 00:14:49.852
letters from people who've taken
calculus and who got As in calculus

00:14:49.852 --> 00:14:52.074
and then they watch that video and
they say, "Oh, I finally understand

00:14:52.074 --> 00:14:56.423
what calculus is about,
what's it even good for."

00:14:57.353 --> 00:15:00.404
So there's that, and I think introduction
to trigonometry is a huge pain point

00:15:00.404 --> 00:15:04.267
for a lot of students and so
a lot of them kind of end up
on Youtube looking for help.

00:15:04.267 --> 00:15:06.120
Brian: So, what's the future for you?

00:15:06.152 --> 00:15:09.988
Is it collaboration with others like
you, if there are any others like you

00:15:09.988 --> 00:15:12.036
to continue to grow or something else?

00:15:12.697 --> 00:15:16.880
Salman: You know I'm very open
to it, and when I started off
when I only had 100 videos

00:15:16.880 --> 00:15:20.507
I thought that was the road I would
take, that I would find a team of other

00:15:20.507 --> 00:15:23.486
people like me and we would
just collectively do it.

00:15:23.486 --> 00:15:26.400
I'm open to it but I've
kind of committed myself.

00:15:26.400 --> 00:15:27.491
I enjoy doing it.

00:15:27.491 --> 00:15:32.071
I've done 1500 videos over four
years and most of those four years

00:15:32.071 --> 00:15:33.786
I was working a day job.

00:15:33.786 --> 00:15:36.200
I've done about 600 since
I quit my job in September.

00:15:36.200 --> 00:15:40.822
So I'm fairly confident that I
could tackle a big chunk of it,

00:15:40.822 --> 00:15:42.710
but I'm sure other people will join in.

00:15:43.535 --> 00:15:47.109
But then augment it with software
and kind of a community built

00:15:47.109 --> 00:15:49.623
around the 200,000 people
using the site right now.

00:15:50.437 --> 00:15:52.735
To really turn it into a virtual school.

00:15:53.404 --> 00:15:55.156
Brian: Hey Salman, great
stuff. Thanks for coming on.

00:15:55.156 --> 00:15:57.000
Salman: Great. Thank you.


Multiplying Polynomials 1

WEBVTT
Kind: captions
Language: en

00:00:01.020 --> 00:00:04.750
This problem says, some people
use the letters F-O-I-L, or

00:00:04.750 --> 00:00:08.480
FOIL, as a strategy for making
sure they have multiplied all

00:00:08.480 --> 00:00:12.390
the terms. Use the FOIL approach
to multiply 3x plus

00:00:12.390 --> 00:00:15.240
2, times 5x minus 7.

00:00:15.240 --> 00:00:18.010
I'm going to let you know just
from the get-go, I personally

00:00:18.010 --> 00:00:21.750
don't like FOIL because it's
kind of just a mnemonic to

00:00:21.750 --> 00:00:23.970
kind of memorize steps, and
you really don't know what

00:00:23.970 --> 00:00:24.470
you're doing.

00:00:24.470 --> 00:00:27.620
But since this problem
says use FOIL, I

00:00:27.620 --> 00:00:29.810
guess we'll use FOIL.

00:00:29.810 --> 00:00:31.250
So let's just use FOIL.

00:00:31.250 --> 00:00:35.310
So FOIL stands for-- and
that's why I hate these

00:00:35.310 --> 00:00:37.370
mnemonics, because as soon as
you're done with the test

00:00:37.370 --> 00:00:39.110
you're going to forget this, and
when you're 30 years old

00:00:39.110 --> 00:00:41.590
you're not going to even
remember what FOIL stands for.

00:00:41.590 --> 00:00:43.710
Especially how to apply it.

00:00:43.710 --> 00:00:53.260
But it stands for first,
outside, inside, and last. And

00:00:53.260 --> 00:00:55.260
the whole point of it is to make
sure that you multiply

00:00:55.260 --> 00:00:58.460
every term in this first
expression with every other

00:00:58.460 --> 00:00:59.710
term in the last expression.

00:00:59.710 --> 00:01:01.400
Now, the reason why you're doing
it is you're really just

00:01:01.400 --> 00:01:03.290
doing the distributive
property twice.

00:01:03.290 --> 00:01:05.570
And I'll do that later in this
video, because I really

00:01:05.570 --> 00:01:08.250
dislike just memorizing FOIL.

00:01:08.250 --> 00:01:10.380
But since we have it written
there, let's do it.

00:01:10.380 --> 00:01:13.500
So first says let's multiply
the two first terms. So

00:01:13.500 --> 00:01:15.720
first-- let me do it in
different colors-- the first

00:01:15.720 --> 00:01:18.760
terms, two first terms
was a 3x and a 5x.

00:01:18.760 --> 00:01:23.810
So you multiply the
3x times the 5x.

00:01:23.810 --> 00:01:26.700
Then the outside says multiply
the outside numbers.

00:01:26.700 --> 00:01:29.470
So multiply the 3x and
the negative 7.

00:01:29.470 --> 00:01:31.720
So notice, we've multiplied the
3x times the 5x, and now

00:01:31.720 --> 00:01:34.180
we're going to multiply the
3x times the negative 7.

00:01:34.180 --> 00:01:38.450
So plus 3x times negative 7.

00:01:38.450 --> 00:01:40.130
That's the outside.

00:01:40.130 --> 00:01:43.370
Then it tells you to
do the inside.

00:01:43.370 --> 00:01:46.570
The inside numbers are
the 2 and the 5x.

00:01:46.570 --> 00:01:49.690
So plus 2 times 5x.

00:01:49.690 --> 00:01:54.160
And then L stands for last, or
the last numbers, or 2 times

00:01:54.160 --> 00:01:55.340
negative 7.

00:01:55.340 --> 00:01:58.690
Plus 2 times negative 7.

00:01:58.690 --> 00:02:00.530
Now we just have to simplify.

00:02:00.530 --> 00:02:07.760
3x times 5x is 15x squared, plus
3x times negative 7 is

00:02:07.760 --> 00:02:10.580
negative 21x.

00:02:10.580 --> 00:02:15.070
2 times 5x is 10x.

00:02:15.070 --> 00:02:20.010
And then you have 2 times
negative 7 is negative 14.

00:02:20.010 --> 00:02:22.990
And now we can combine these
middle two terms right here.

00:02:22.990 --> 00:02:26.010
If I have negative 21x and then
I'm adding 10x to that,

00:02:26.010 --> 00:02:28.510
that takes me back
to negative 11x.

00:02:28.510 --> 00:02:31.290
So this is negative 11x.

00:02:31.290 --> 00:02:33.810
And we have the 15x
squared out front.

00:02:33.810 --> 00:02:40.580
15x squared minus
11x, minus 14.

00:02:40.580 --> 00:02:43.490
Which is the product of
these two expressions.

00:02:43.490 --> 00:02:45.790
So FOIL does give you the right
answer, but it's just

00:02:45.790 --> 00:02:48.450
not always obvious why you're
doing all of these different

00:02:48.450 --> 00:02:49.890
combinations of multiplications.

00:02:49.890 --> 00:02:51.370
But this is the right answer.

00:02:51.370 --> 00:02:53.910
But what I want to do is take
a side here to just show you

00:02:53.910 --> 00:02:56.760
that as long as you just knew
the distributive property, you

00:02:56.760 --> 00:02:57.770
could have done this problem.

00:02:57.770 --> 00:02:59.970
You did not have to know FOIL.

00:02:59.970 --> 00:03:01.460
And this way you'll
never forget it.

00:03:01.460 --> 00:03:11.200
So if I'm going to multiply 3x
plus 2, times 5x minus 7, what

00:03:11.200 --> 00:03:14.280
you can think about is, this is
just some number here, and

00:03:14.280 --> 00:03:16.330
if I'm multiplying times this
expression, I can just

00:03:16.330 --> 00:03:18.050
distribute it onto
this expression.

00:03:18.050 --> 00:03:20.430
If this was just a, I don't
know, if this was just a 2, we

00:03:20.430 --> 00:03:23.220
would say, OK, this is the same
thing as 2 times 5x, plus

00:03:23.220 --> 00:03:24.490
2 times negative 7.

00:03:24.490 --> 00:03:26.010
That's what the distributive
property is.

00:03:26.010 --> 00:03:27.730
So let's just distribute
it out.

00:03:27.730 --> 00:03:28.220
Let's just distribute.

00:03:28.220 --> 00:03:32.320
We have to just multiply this
whole thing times each of

00:03:32.320 --> 00:03:34.920
those terms. That's just the
distributive property.

00:03:34.920 --> 00:03:41.030
So you get 3x plus
2, times 5x.

00:03:41.030 --> 00:03:44.120
I could write the 5x
out front here.

00:03:44.120 --> 00:03:45.950
So that's just that
times that.

00:03:45.950 --> 00:03:53.120
And then you're going to have
plus 3x plus 2, times the

00:03:53.120 --> 00:03:54.760
second term, times negative 7.

00:03:54.760 --> 00:03:56.760
Now, I could just put a negative
plus here and a

00:03:56.760 --> 00:03:57.030
negative 7.

00:03:57.030 --> 00:04:00.030
Or I could say it's negative
7 times 3x, plus 2.

00:04:00.030 --> 00:04:04.020
So it's 5x times 3x plus 2,
minus 7 times 3x plus 2.

00:04:04.020 --> 00:04:06.510
This was just the distributive
property.

00:04:06.510 --> 00:04:09.910
Now, again, we can apply the
distributive property to get

00:04:09.910 --> 00:04:14.000
the 5x times each of these
terms. So this is going to be

00:04:14.000 --> 00:04:22.130
5x times 3x, plus 5x, times 2.

00:04:26.690 --> 00:04:28.940
Just doing the distributive
property on this.

00:04:28.940 --> 00:04:32.340
Now, we do it again here, so
that we have minus 7 or

00:04:32.340 --> 00:04:33.590
negative 7 times 3x.

00:04:39.750 --> 00:04:42.290
And then negative 7 times 2.

00:04:42.290 --> 00:04:46.380
So minus 7 times 2.

00:04:46.380 --> 00:04:47.610
And what do we get?

00:04:47.610 --> 00:04:49.360
Scroll down a little bit.

00:04:49.360 --> 00:04:51.330
I'll do these in a
neutral color.

00:04:51.330 --> 00:04:55.410
We get 5x times 3x
is 15x squared.

00:04:55.410 --> 00:04:58.350
5x times 2 is 10x.

00:04:58.350 --> 00:05:02.620
7 times 3x is minus 21x.

00:05:02.620 --> 00:05:06.000
And then negative 7 times
2 is minus 14.

00:05:06.000 --> 00:05:10.790
And just like we saw up here,
these two terms can be merged

00:05:10.790 --> 00:05:14.530
into 10x minus 21x
is negative 11x.

00:05:14.530 --> 00:05:18.890
So we get 15x squared
minus 11x, minus 14.

00:05:18.890 --> 00:05:21.060
Could have done it just doing
the distributive property

00:05:21.060 --> 00:05:23.510
twice, which you're less likely
to forget when you're

00:05:23.510 --> 00:05:24.540
30 years old.

00:05:24.540 --> 00:05:27.400
Or if you like, you
can memorize FOIL.


Applying radical equations 3

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:03.940
We're told that the radius of a
sphere can be measured using

00:00:03.940 --> 00:00:08.760
the formula r is equal to the
cube root of 3 times v,

00:00:08.760 --> 00:00:12.970
divided by 4pi, where v is
equal to the volume.

00:00:12.970 --> 00:00:13.660
That's interesting.

00:00:13.660 --> 00:00:15.760
Normally you have it expressed
the other way, volume as a

00:00:15.760 --> 00:00:17.230
function of radius.

00:00:17.230 --> 00:00:20.120
Here they have radius as
a function of volume.

00:00:20.120 --> 00:00:23.680
If the radius of a sphere is 6
inches-- so they're giving us

00:00:23.680 --> 00:00:25.730
the radius; they're saying
it's 6 inches.

00:00:25.730 --> 00:00:27.010
So they're giving us that right

00:00:27.010 --> 00:00:29.460
there-- what is its volume?

00:00:29.460 --> 00:00:33.180
So we have 6 inches--
everything's in inches-- is

00:00:33.180 --> 00:00:40.820
equal to the cube root of 3
times the volume-- that's what

00:00:40.820 --> 00:00:44.000
we need to solve for--
divided by 4pi.

00:00:44.000 --> 00:00:47.240
A good first step to get rid of
this cube root, is take the

00:00:47.240 --> 00:00:50.620
cube of both sides or raise both
sides of this equation to

00:00:50.620 --> 00:00:51.820
the third power.

00:00:51.820 --> 00:00:54.630
Let's see, it's 6 inches, you
raise it to the third power.

00:00:54.630 --> 00:00:56.850
You have the cube root
of 3v over 4pi.

00:00:56.850 --> 00:00:58.810
Raise it to the third power.

00:00:58.810 --> 00:01:01.830
And so the left-hand side of
this equation becomes what?

00:01:01.830 --> 00:01:04.200
6 times 6 is 36.

00:01:04.200 --> 00:01:10.610
36 times 6 is 180,
plus 36 is 216.

00:01:10.610 --> 00:01:12.620
So you have 216.

00:01:12.620 --> 00:01:14.603
And you had 6 inches here-- now
that you're cubing it, it

00:01:14.603 --> 00:01:17.770
would actually now become 216
cubic inches, or inches

00:01:17.770 --> 00:01:24.630
cubed-- is equal to-- if you
take the 3v divided by 4pi,

00:01:24.630 --> 00:01:26.480
the cube root is essentially
taking it to the 1/3 power.

00:01:26.480 --> 00:01:27.720
Then take that to
the third power.

00:01:27.720 --> 00:01:29.260
They unwind each other.

00:01:29.260 --> 00:01:33.600
So this just becomes
3v over 4pi.

00:01:33.600 --> 00:01:36.670
Now, to solve for volume, we can
just multiply both sides

00:01:36.670 --> 00:01:41.500
of this equation times the
inverse of its coefficient,

00:01:41.500 --> 00:01:44.660
times 4pi over 3.

00:01:44.660 --> 00:01:49.840
Times-- let me scroll to the
leftt-- times 4pi over 3.

00:01:49.840 --> 00:01:52.160
Let me make sure you can
see what I'm writing.

00:01:52.160 --> 00:01:54.240
And the whole reason why I'm
multiplying by 4pi over 3 is

00:01:54.240 --> 00:01:56.960
so that it cancels out
with the 3 over 4pi.

00:01:56.960 --> 00:02:00.700
And you have your 4pi cancels
out with a 4pi, and then you

00:02:00.700 --> 00:02:03.330
have your 3 canceling
out with a 3.

00:02:03.330 --> 00:02:08.160
And then you have, your volume
is going to be equal to 4pi

00:02:08.160 --> 00:02:11.630
times 216 over 3.

00:02:11.630 --> 00:02:13.050
We could write it this way.

00:02:13.050 --> 00:02:24.070
216 times 4pi over 3.

00:02:24.070 --> 00:02:25.350
And we could simplify this.

00:02:25.350 --> 00:02:29.360
216, the digits add up to--
2 plus 1, plus 6 is 9.

00:02:29.360 --> 00:02:30.610
So it will be divisible by 3.

00:02:30.610 --> 00:02:32.880
That's a fast way to test
whether something's

00:02:32.880 --> 00:02:34.350
divisible by 3.

00:02:34.350 --> 00:02:35.450
So let's see what it is.

00:02:35.450 --> 00:02:39.240
3 goes into 216.

00:02:39.240 --> 00:02:41.110
3 goes into 21 seven times.

00:02:41.110 --> 00:02:43.290
7 times 3 is 21.

00:02:43.290 --> 00:02:44.370
Bring down the 6.

00:02:44.370 --> 00:02:45.890
You have no remainder,
so bring down the 6.

00:02:45.890 --> 00:02:48.260
3 goes into 6 two times.

00:02:48.260 --> 00:02:50.070
2 times 3 is 6.

00:02:50.070 --> 00:02:52.300
No remainder, so it's 72.

00:02:52.300 --> 00:02:54.950
So 216 divided by 3 is 72.

00:02:54.950 --> 00:02:58.210
So it's 72 times 4pi is
equal to the volume.

00:02:58.210 --> 00:03:00.120
It's going to be cubic inches.

00:03:00.120 --> 00:03:02.690
So 4 times 72.

00:03:02.690 --> 00:03:07.140
4 times 70 is 280.

00:03:07.140 --> 00:03:09.430
And then you have 4 times
2, which is 8.

00:03:09.430 --> 00:03:14.170
So it'll be 288pi
cubic inches.

00:03:14.170 --> 00:03:16.870
That's our volume.

00:03:16.870 --> 00:03:18.460
And we're done.


Fractional exponent expressions 1

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:05.900
We're asked to simplify d times
4d to the 1/2 power.

00:00:05.900 --> 00:00:07.000
So let's see what
we can do here.

00:00:07.000 --> 00:00:09.540
And I think there's multiple
ways to kind of simplify it,

00:00:09.540 --> 00:00:11.260
and we'll explore all of them.

00:00:11.260 --> 00:00:13.150
So d times 4d to the 1/2.

00:00:13.150 --> 00:00:17.060
4d to the 1/2, let me just focus
on that right there.

00:00:17.060 --> 00:00:18.490
4d to the 1/2.

00:00:18.490 --> 00:00:21.360
If you take the product of two
things and you raise that to

00:00:21.360 --> 00:00:25.110
an exponent, that's the same
thing as taking each of those

00:00:25.110 --> 00:00:27.950
terms to that exponent and
then taking the product.

00:00:27.950 --> 00:00:33.210
So 4d to the 1/2 is the same
thing as 4 to the 1/2 times d

00:00:33.210 --> 00:00:34.260
to the 1/2.

00:00:34.260 --> 00:00:39.190
And of course, you have
this d out front.

00:00:39.190 --> 00:00:40.670
And I could put the parentheses
here, but it

00:00:40.670 --> 00:00:41.730
doesn't matter at this point.

00:00:41.730 --> 00:00:43.250
Now you just have three
things that you're

00:00:43.250 --> 00:00:44.590
taking the product of.

00:00:44.590 --> 00:00:47.310
So I'll put this d out
front just like that.

00:00:47.310 --> 00:00:49.230
And now there's a couple of ways
you could think about it.

00:00:49.230 --> 00:00:52.130
You have this 4 to the 1/2 power
and you're taking its

00:00:52.130 --> 00:00:54.650
principal square root, so its
positive square root.

00:00:54.650 --> 00:00:57.030
So that right there
is going to be 2.

00:00:57.030 --> 00:00:59.780
So 4 to the 1/2, that
right there is going

00:00:59.780 --> 00:01:02.790
to be equal to 2.

00:01:02.790 --> 00:01:05.180
And then, there's a couple of
ways you could think about it.

00:01:05.180 --> 00:01:08.740
You have this d out front.

00:01:08.740 --> 00:01:09.990
And then you have
a d to the 1/2.

00:01:12.660 --> 00:01:15.440
And there's several ways that
you can now represent this,

00:01:15.440 --> 00:01:18.640
depending on what you feel--
I don't know--

00:01:18.640 --> 00:01:20.120
aesthetically pleasant.

00:01:20.120 --> 00:01:23.110
You could say, well look, this
is d to the first power.

00:01:23.110 --> 00:01:24.590
This is d to the 1/2 power.

00:01:24.590 --> 00:01:27.580
I could add them.

00:01:27.580 --> 00:01:32.330
So one way to say it is it's 2
times d to the 1 plus 1/2.

00:01:32.330 --> 00:01:34.630
You can either say 1 and
1/2 power or 3/2

00:01:34.630 --> 00:01:36.110
power if you want.

00:01:36.110 --> 00:01:38.810
So that's one simplification
of this expression.

00:01:38.810 --> 00:01:41.470
Another simplification would be
well, you could say, look.

00:01:41.470 --> 00:01:46.280
d to the 1/2, that is the same
thing as the square root of d.

00:01:46.280 --> 00:01:51.650
So you could say this is the
same thing as 2 times d times

00:01:51.650 --> 00:01:53.670
the square root of d.

00:01:53.670 --> 00:01:56.570
So any of these are actual
simplifications, but it's

00:01:56.570 --> 00:01:58.030
interesting to see that
they're really all

00:01:58.030 --> 00:02:00.800
representing the same thing.


Example 3: Modeling with polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:03.120
We're told that the glass
portion of the window shown

00:00:03.120 --> 00:00:05.720
below-- that's the window--
has a height to width the

00:00:05.720 --> 00:00:07.660
ratio of 3:2.

00:00:07.660 --> 00:00:10.450
So the height of the glass
portion can be represented as

00:00:10.450 --> 00:00:21.250
3x and the width of the
glass portion can be

00:00:21.250 --> 00:00:22.500
represented as 2x.

00:00:27.340 --> 00:00:31.890
The wood trim adds 7 inches to
the total width and 8 inches

00:00:31.890 --> 00:00:33.540
to the total height.

00:00:33.540 --> 00:00:34.670
So let me write that down.

00:00:34.670 --> 00:00:37.450
So 8 in height, we're
doing it in yellow.

00:00:37.450 --> 00:00:40.650
So 8 inches to the total height,
I'll do it in yellow,

00:00:40.650 --> 00:00:44.330
and 7 inches to the total
width, I'll do in this

00:00:44.330 --> 00:00:45.680
purplish color, right?

00:00:45.680 --> 00:00:47.670
We're doing the width
in purple.

00:00:47.670 --> 00:00:50.050
Now what they say is write a
polynomial expression that

00:00:50.050 --> 00:00:55.350
represents the total area of
the window, including the

00:00:55.350 --> 00:00:56.450
glass and wood.

00:00:56.450 --> 00:00:57.750
Simplify the expression.

00:00:57.750 --> 00:01:00.480
Now, the way they've described
it, they're telling us the

00:01:00.480 --> 00:01:02.460
ratio of 3:2 for the
glass portion.

00:01:02.460 --> 00:01:04.500
I think when they're talking
about the glass portion,

00:01:04.500 --> 00:01:06.610
they're talking about this
entire portion right here.

00:01:09.130 --> 00:01:11.370
Even though it's not all glass,
you have that thing in

00:01:11.370 --> 00:01:16.010
the middle, you have this
divider right over here,

00:01:16.010 --> 00:01:18.450
that's not glass, but I think
they're including that as the

00:01:18.450 --> 00:01:20.160
glass portion, because they're
not giving us any

00:01:20.160 --> 00:01:21.460
measurements for that.

00:01:21.460 --> 00:01:25.125
So the height of the glass
portion can be

00:01:25.125 --> 00:01:26.830
represented as 3x.

00:01:26.830 --> 00:01:31.050
So this height right here of
just the glass portion is 3x.

00:01:31.050 --> 00:01:31.920
You can't see that.

00:01:31.920 --> 00:01:34.360
Let me do it in a
better color.

00:01:34.360 --> 00:01:38.510
This portion right here is 3x,
just this distance right here.

00:01:38.510 --> 00:01:41.290
That is the height of
the glass portion.

00:01:41.290 --> 00:01:44.610
And then they tell us the wood
trim adds 8 inches to the

00:01:44.610 --> 00:01:45.650
total height.

00:01:45.650 --> 00:01:51.670
So if you take that 3x and then
you add that plus that,

00:01:51.670 --> 00:01:55.600
this whole distance over here,
which is the glass portion

00:01:55.600 --> 00:02:00.740
plus the wood trim on either
end, it adds 8 inches to the

00:02:00.740 --> 00:02:01.310
glass portion.

00:02:01.310 --> 00:02:04.990
The glass portion is 3x, so
we're going to add 8 to it.

00:02:04.990 --> 00:02:07.990
So this height right
here 3x plus 8.

00:02:07.990 --> 00:02:10.090
And now let's think
about the width.

00:02:10.090 --> 00:02:13.090
The width of the glass portion
can be represented as 2x, so

00:02:13.090 --> 00:02:17.020
that distance right there, just
the glass portion, is 2x.

00:02:17.020 --> 00:02:20.255
But then they say when you add
the wood trim, the wood trim

00:02:20.255 --> 00:02:23.550
adds 7 inches to the total
width, so this total width

00:02:23.550 --> 00:02:27.770
right here, because we're adding
that piece and that

00:02:27.770 --> 00:02:30.050
piece to it, is 2x plus 7.

00:02:33.130 --> 00:02:34.860
That is the total width of
the window, the total

00:02:34.860 --> 00:02:36.670
height is 3x plus 8.

00:02:36.670 --> 00:02:38.460
Now, what they want us to
do is write a polynomial

00:02:38.460 --> 00:02:40.900
expression that represents the
total area of the window

00:02:40.900 --> 00:02:42.510
including the glass and wood.

00:02:42.510 --> 00:02:44.230
Simplify the expression.

00:02:44.230 --> 00:02:45.890
Now, the total area is
just going to be the

00:02:45.890 --> 00:02:47.630
height times the width.

00:02:47.630 --> 00:02:48.950
So let's just write that down.

00:02:48.950 --> 00:02:53.360
The area is just going to be
equal to the height, which is

00:02:53.360 --> 00:02:58.996
3x plus 8 times the width,
times 2x plus 7.

00:03:03.270 --> 00:03:05.700
And that is the expression
we've written up.

00:03:05.700 --> 00:03:06.680
Well, we haven't done
it as a polynomial.

00:03:06.680 --> 00:03:09.300
We've written it as a product
of two binomials.

00:03:09.300 --> 00:03:12.580
And if we want to simplify it,
we can just multiply it out.

00:03:12.580 --> 00:03:15.560
So the way you can think about
it is we multiply-- you're

00:03:15.560 --> 00:03:17.760
doing the distributive property
twice essentially.

00:03:17.760 --> 00:03:21.150
But you could say let's multiply
3x times 2x, so you

00:03:21.150 --> 00:03:23.680
get 3x times 2x.

00:03:26.390 --> 00:03:28.160
That's multiplying
it times that.

00:03:28.160 --> 00:03:35.600
Then you want to multiply 3x
times 7, which is 21x-- let me

00:03:35.600 --> 00:03:42.260
just write it this way--
plus 3x times 7.

00:03:42.260 --> 00:03:44.740
And then you want to multiply
this 8 times both of these

00:03:44.740 --> 00:03:50.320
terms. So then you have
plus 8 times 2x.

00:03:50.320 --> 00:03:50.900
Let me do this.

00:03:50.900 --> 00:03:59.185
So plus 8 times 2x and then
you have plus 8 times 7.

00:04:05.880 --> 00:04:07.610
And now we just have to
simplify everything.

00:04:07.610 --> 00:04:09.130
So what does this become?

00:04:09.130 --> 00:04:17.920
3x times 2x is 6x squared plus
3x times 7 is plus 21x plus 8

00:04:17.920 --> 00:04:23.700
times 2x, so plus 16x, and then
plus 8 times 7, 8 times 7

00:04:23.700 --> 00:04:26.340
is 56, plus 56.

00:04:26.340 --> 00:04:28.370
We can combine these
two middle terms.

00:04:28.370 --> 00:04:30.650
21x plus 16x is what?

00:04:30.650 --> 00:04:32.120
37x.

00:04:32.120 --> 00:04:34.940
So then we get-- let me just
do it in another color-- 6x

00:04:34.940 --> 00:04:40.870
squared plus 37x plus 56.

00:04:40.870 --> 00:04:46.260
Now we've expressed it as
a simplified polynomial

00:04:46.260 --> 00:04:47.510
expression.


Adding and subtracting polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:05.240
Simplify 16x plus 14 minus
this entire quantity 3x

00:00:05.240 --> 00:00:07.680
squared plus x minus 9.

00:00:07.680 --> 00:00:09.910
So let me just rewrite it and
I'll get rid of some of the

00:00:09.910 --> 00:00:10.906
parentheses if we can.

00:00:10.906 --> 00:00:13.360
So we have this 16x
right here.

00:00:13.360 --> 00:00:18.170
16x plus 14.

00:00:18.170 --> 00:00:20.110
we don't have to have those
parentheses, it doesn't change

00:00:20.110 --> 00:00:21.630
the order of operations.

00:00:21.630 --> 00:00:24.080
And then we can subtract
from that.

00:00:24.080 --> 00:00:26.150
Well, we can just distribute
this negative sign.

00:00:26.150 --> 00:00:28.020
So instead of having a positive
3x squared, if we

00:00:28.020 --> 00:00:29.390
multiply it times
this negative 1,

00:00:29.390 --> 00:00:32.240
it is minus 3x squared.

00:00:32.240 --> 00:00:34.530
That's that term right
over there.

00:00:34.530 --> 00:00:36.860
And then you have this
x over here.

00:00:36.860 --> 00:00:39.490
It's a positive x, but it's
being multiplied by this

00:00:39.490 --> 00:00:40.510
negative 1 out here.

00:00:40.510 --> 00:00:42.900
So it is minus x.

00:00:42.900 --> 00:00:46.600
And then finally, you have this
negative 9 out here, but

00:00:46.600 --> 00:00:49.290
it's being multiplied by a
negative 1 out in front of the

00:00:49.290 --> 00:00:50.530
parentheses, so a
negative times a

00:00:50.530 --> 00:00:52.040
negative is a positive.

00:00:52.040 --> 00:00:53.750
So plus 9.

00:00:53.750 --> 00:00:56.930
And so now we can just combine
terms that have the same

00:00:56.930 --> 00:00:58.460
degree of x in them.

00:00:58.460 --> 00:01:00.180
And I'll start with the
highest degree term.

00:01:00.180 --> 00:01:01.580
The highest degree
term right here.

00:01:01.580 --> 00:01:03.510
And the degree just
means x raised

00:01:03.510 --> 00:01:05.030
to the highest exponent.

00:01:05.030 --> 00:01:07.670
The highest degree term here
is negative 3x squared.

00:01:07.670 --> 00:01:09.600
It's the second degree term.

00:01:09.600 --> 00:01:11.930
x is being raised to
the second power.

00:01:11.930 --> 00:01:14.650
So I'll just write here,
negative 3x squared.

00:01:14.650 --> 00:01:16.130
It's the only when we have.

00:01:16.130 --> 00:01:20.550
Then we have the x terms.
We have 16x minus x.

00:01:20.550 --> 00:01:23.200
So if you have 16 of something
and you subtract 1 away from

00:01:23.200 --> 00:01:25.180
it, you're going to have 15x.

00:01:25.180 --> 00:01:28.090
So plus 15x.

00:01:28.090 --> 00:01:31.000
And then finally, you have your
constant terms. 14 and

00:01:31.000 --> 00:01:32.200
then you have plus 9.

00:01:32.200 --> 00:01:34.870
So 14 plus 9 is 23.

00:01:34.870 --> 00:01:36.480
Plus 23.

00:01:36.480 --> 00:01:38.200
And we are all done.

00:01:38.200 --> 00:01:41.380
We've simplified
the polynomial.


Example 4: Adding and subtracting polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:04.013
We're asked to simplify this
huge, long expression here. x

00:00:04.013 --> 00:00:07.720
to the third plus 3x minus 6--
that's in parentheses-- plus

00:00:07.720 --> 00:00:10.700
negative 2x squared
plus x minus 2.

00:00:10.700 --> 00:00:13.420
And then minus the quantity
3x minus 4.

00:00:13.420 --> 00:00:15.790
So a good place to start, we'll
just rewrite this and

00:00:15.790 --> 00:00:18.550
see if we can eliminate the
parentheses in this step.

00:00:18.550 --> 00:00:20.420
So let's just start
at the beginning.

00:00:20.420 --> 00:00:24.270
We have the x to the third
right over there.

00:00:24.270 --> 00:00:28.435
So x to the third and then
plus 3x-- I'll do that in

00:00:28.435 --> 00:00:29.685
pink-- plus 3x.

00:00:32.090 --> 00:00:36.510
And then we have a minus 6.

00:00:36.510 --> 00:00:38.570
And we don't have to put the
parentheses around there,

00:00:38.570 --> 00:00:40.770
those don't really
change anything.

00:00:40.770 --> 00:00:42.750
And we don't have to even write
these-- do anything with

00:00:42.750 --> 00:00:43.480
these parentheses.

00:00:43.480 --> 00:00:44.225
We can eliminate them.

00:00:44.225 --> 00:00:45.760
Just because there's a positive
sign out here we

00:00:45.760 --> 00:00:47.130
don't have to distribute
anything.

00:00:47.130 --> 00:00:48.670
Distributing a positive
sign doesn't do

00:00:48.670 --> 00:00:49.980
anything to these numbers.

00:00:49.980 --> 00:00:53.490
So then plus, we have a
negative 2x squared.

00:00:53.490 --> 00:00:56.600
So this term right here
is negative 2x

00:00:56.600 --> 00:00:58.440
squared, or minus x squared.

00:00:58.440 --> 00:01:00.590
And then we have a plus x.

00:01:00.590 --> 00:01:01.890
We have a plus x.

00:01:04.940 --> 00:01:06.435
Then we have a minus 2.

00:01:09.340 --> 00:01:12.180
Then we have a negative sign
times this whole expression.

00:01:12.180 --> 00:01:13.730
So we're going to have to
distribute the negative sign.

00:01:13.730 --> 00:01:16.190
So it's a positive 3x, but
it's being multiplied by

00:01:16.190 --> 00:01:17.250
negative 1.

00:01:17.250 --> 00:01:19.480
So it's really a negative 3x.

00:01:19.480 --> 00:01:25.040
So minus 3x, then you have a
negative-- you can imagine

00:01:25.040 --> 00:01:27.240
this is a negative 1 implicitly
out here-- negative

00:01:27.240 --> 00:01:28.090
1 times negative 4.

00:01:28.090 --> 00:01:29.570
That's a positive 4.

00:01:29.570 --> 00:01:30.820
So plus 4.

00:01:33.410 --> 00:01:36.020
Now, we could combine
terms of similar

00:01:36.020 --> 00:01:37.260
degree, of the same degree.

00:01:37.260 --> 00:01:39.360
Now, first we have an x to the
third term and I think it's

00:01:39.360 --> 00:01:42.150
the only third degree term here,
because we have x being

00:01:42.150 --> 00:01:43.760
raised to the third power.

00:01:43.760 --> 00:01:44.770
So let me just rewrite
it here.

00:01:44.770 --> 00:01:45.835
We have x to the third.

00:01:45.835 --> 00:01:48.400
And now let's look at our x
squared terms. Looks like we

00:01:48.400 --> 00:01:49.680
only have one.

00:01:49.680 --> 00:01:51.340
We only have this
term right here.

00:01:51.340 --> 00:01:54.740
So we have minus 2x squared.

00:01:54.740 --> 00:01:56.130
And then what about
our x terms?

00:01:56.130 --> 00:02:00.980
We have a 3x plus an
x minus a 3x again.

00:02:00.980 --> 00:02:04.143
So that 3x minus the 3x would
cancel out, and you're just

00:02:04.143 --> 00:02:05.490
left with an x.

00:02:05.490 --> 00:02:07.490
So plus x.

00:02:07.490 --> 00:02:10.620
And then finally our constant
terms. Negative 6

00:02:10.620 --> 00:02:12.810
minus 2 plus 4.

00:02:12.810 --> 00:02:15.790
Negative 6 minus 2 gets
us to negative 8.

00:02:15.790 --> 00:02:20.150
Plus 4 is negative 4.

00:02:20.150 --> 00:02:20.850
And we are done.

00:02:20.850 --> 00:02:22.560
We have simplified
the expression.

00:02:22.560 --> 00:02:24.570
Now we just have a four
term polynomial.


Example 3: Multiplying a binomial by a polynomial word problem

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:05.110
Find the volume of a tank whose
base has an area of 3x

00:00:05.110 --> 00:00:08.860
squared plus 30x plus 5
square feet and whose

00:00:08.860 --> 00:00:10.690
height is 8x minus 5.

00:00:10.690 --> 00:00:13.880
So let me draw a potential
tank here.

00:00:13.880 --> 00:00:17.150
Maybe it's a cylinder, so let
me draw a cylindrical tank

00:00:17.150 --> 00:00:21.160
here, just like that.

00:00:21.160 --> 00:00:23.260
It's supposed to be a cylinder,
and maybe if it was

00:00:23.260 --> 00:00:25.860
transparent, you would see this
back side over there.

00:00:25.860 --> 00:00:26.980
And these two are the same.

00:00:26.980 --> 00:00:28.850
These are supposed to
be straight lines.

00:00:28.850 --> 00:00:32.659
And the volume of a-- let me
just label it first. So the

00:00:32.659 --> 00:00:35.130
area of the base, which is the
same as the area of the top

00:00:35.130 --> 00:00:40.690
here or of the base here, the
area of that is 3x squared

00:00:40.690 --> 00:00:43.330
plus 30x plus 5.

00:00:43.330 --> 00:00:46.920
And they tell us the height
is 8x minus 5.

00:00:46.920 --> 00:00:50.600
The height of this tank
is 8x minus 5.

00:00:54.890 --> 00:00:59.080
And if you want to find the
volume of a three-dimensional

00:00:59.080 --> 00:01:01.540
object like this, you just
multiply the area of the base

00:01:01.540 --> 00:01:02.720
times the height.

00:01:02.720 --> 00:01:07.350
So the volume is going to be the
area of the base, which is

00:01:07.350 --> 00:01:13.500
3x squared plus 30x plus
5, times the height,

00:01:13.500 --> 00:01:15.440
times 8x minus 5.

00:01:18.750 --> 00:01:22.030
Now, to multiply something like
this out, it might seem

00:01:22.030 --> 00:01:24.550
really complicated, but we once
again, we just have to do

00:01:24.550 --> 00:01:26.070
the distributive property.

00:01:26.070 --> 00:01:28.720
If you viewed this big thing in
pink here as just a number,

00:01:28.720 --> 00:01:31.890
if this was like the number 7,
you'd just say, well, this is

00:01:31.890 --> 00:01:37.430
going to be 7 times 8x minus 7
times 5 or minus 5 times 7.

00:01:37.430 --> 00:01:38.580
You would just distribute
it out.

00:01:38.580 --> 00:01:42.400
You would just multiply this
entire thing times each of the

00:01:42.400 --> 00:01:44.510
terms. That's what the
distributive property tells us

00:01:44.510 --> 00:01:45.480
when you first learned it.

00:01:45.480 --> 00:01:46.410
So let's do that.

00:01:46.410 --> 00:01:49.940
So it's going to be this entire
thing times 8x, or we

00:01:49.940 --> 00:01:52.440
can view it as 8x times
this entire thing.

00:01:52.440 --> 00:02:05.600
So 8x times this entire thing:
3x squared plus 30x plus 5,

00:02:05.600 --> 00:02:10.979
minus 5 times the entire thing
again, or the entire thing

00:02:10.979 --> 00:02:12.670
times minus 5.

00:02:12.670 --> 00:02:19.360
So once again you get 3x squared
plus 30x plus 5.

00:02:19.360 --> 00:02:20.840
And now we just multiply
these out.

00:02:20.840 --> 00:02:23.410
We just distribute out the 8x
over this whole thing and we

00:02:23.410 --> 00:02:26.940
distribute out the negative
5 over that whole thing.

00:02:26.940 --> 00:02:32.050
So then we get 8x times
3x squared is 24x

00:02:32.050 --> 00:02:33.570
to the third power.

00:02:33.570 --> 00:02:35.570
8x times 30x is what?

00:02:35.570 --> 00:02:41.060
That's 240x squared, so
plus 240x squared.

00:02:41.060 --> 00:02:45.620
8x times 5 is plus 40x.

00:02:45.620 --> 00:02:48.490
And then we multiply this
negative 5 out.

00:02:48.490 --> 00:02:57.020
Negative 5 times 3x squared
is negative 15x squared.

00:02:57.020 --> 00:03:02.030
Negative 5 times 30x
is negative 150x.

00:03:02.030 --> 00:03:11.460
And then negative 5 times
5 is negative 25.

00:03:11.460 --> 00:03:13.450
And then we just have to
simplify it from here.

00:03:13.450 --> 00:03:16.220
So we only have one third-degree
term, one thing

00:03:16.220 --> 00:03:17.550
that has an x to the
third in it.

00:03:17.550 --> 00:03:19.780
We have this term right here,
so we'll write that

00:03:19.780 --> 00:03:21.810
as 24x to the third.

00:03:21.810 --> 00:03:23.950
And then what are our
x squared terms?

00:03:23.950 --> 00:03:28.510
We have 240x squared
minus 15x squared.

00:03:28.510 --> 00:03:31.020
So what's two 240 minus 15?

00:03:31.020 --> 00:03:32.960
It's 225x squared.

00:03:32.960 --> 00:03:36.240
So plus 225x squared.

00:03:36.240 --> 00:03:39.940
That's adding that term to that
term right over there.

00:03:39.940 --> 00:03:47.100
And then we have
40x minus 150x.

00:03:47.100 --> 00:03:52.300
That would be negative 110x.

00:03:52.300 --> 00:03:57.330
And then finally we have just
this negative 25 out here.

00:03:57.330 --> 00:03:59.240
That's the only constant term.

00:03:59.240 --> 00:04:00.240
And we're done!

00:04:00.240 --> 00:04:02.420
We found the volume
of the tank.

00:04:02.420 --> 00:04:06.010
It's given by this polynomial
expression right here.

00:04:06.010 --> 00:04:10.540
So this right here is the
volume of the tank.

00:04:10.540 --> 00:04:15.090
It's equal to 24x to the
third plus 225x squared

00:04:15.090 --> 00:04:18.680
minus 110x minus 25.


Special Products of Polynomials 2

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:06.070
Find the area of a square
with side 6x minus 5y.

00:00:06.070 --> 00:00:09.380
So if you have a square, all the
sides are going to be the

00:00:09.380 --> 00:00:10.720
same length, and they're
telling us the

00:00:10.720 --> 00:00:14.400
length is 6x minus 5y.

00:00:14.400 --> 00:00:18.130
The height will also
be 6x minus 5y.

00:00:18.130 --> 00:00:21.090
And if you wanted to find the
area of the square, you would

00:00:21.090 --> 00:00:23.830
just multiply the width times
the height, or you would

00:00:23.830 --> 00:00:30.250
multiply 6x minus 5y
times 6x minus 5y.

00:00:30.250 --> 00:00:32.820
Or another way to think about
it is you're taking 6x minus

00:00:32.820 --> 00:00:38.920
5y to the second power,
or you're

00:00:38.920 --> 00:00:41.920
taking 6x minus 5y squared.

00:00:41.920 --> 00:00:46.960
That's why raising something to
the second power is called

00:00:46.960 --> 00:00:49.150
squaring it because you're
essentially finding the area

00:00:49.150 --> 00:00:50.380
of a square.

00:00:50.380 --> 00:00:52.310
You're going from one dimension
to two dimensions.

00:00:52.310 --> 00:00:55.130
That's where the term
squared comes from.

00:00:55.130 --> 00:00:57.180
Now, when you square something,
you can just do the

00:00:57.180 --> 00:00:59.280
distributive property
twice like we've

00:00:59.280 --> 00:01:00.540
done in several videos.

00:01:00.540 --> 00:01:03.570
But a shortcut when you square
something, and it all just

00:01:03.570 --> 00:01:05.560
comes from working it out by
doing the distributive

00:01:05.560 --> 00:01:08.545
property twice, is you can
square the first term, so

00:01:08.545 --> 00:01:13.170
it'll be 6x squared, and then
you can take the product of

00:01:13.170 --> 00:01:18.660
both terms and multiply it by
2, so times 2 times 6x times

00:01:18.660 --> 00:01:20.230
negative 5y.

00:01:20.230 --> 00:01:23.140
And then you can square
the last term.

00:01:23.140 --> 00:01:24.990
Let me do it in a new color.

00:01:24.990 --> 00:01:29.000
So plus negative 5y squared.

00:01:29.000 --> 00:01:30.580
So that's the last
term right there.

00:01:30.580 --> 00:01:31.730
This is the product
of both of them.

00:01:31.730 --> 00:01:35.100
You multiply it by 2 and that
will be this quantity squared.

00:01:35.100 --> 00:01:37.380
And you can get there by just
doing the distributive

00:01:37.380 --> 00:01:40.045
property twice, or FOIL, however
you want to do it, but

00:01:40.045 --> 00:01:43.010
this is a quicker way so that
when you see a squared, you

00:01:43.010 --> 00:01:44.400
can immediately get to this.

00:01:44.400 --> 00:01:47.650
And just working it out,
you get 6x squared is

00:01:47.650 --> 00:01:49.730
36x squared, right?

00:01:49.730 --> 00:01:52.160
6 squared times x squared.

00:01:52.160 --> 00:01:56.790
And then you have a negative--
let's see, 2 times 6 times 5.

00:01:56.790 --> 00:02:02.730
That is negative 10 times
6, so it's negative 60.

00:02:02.730 --> 00:02:09.280
And then we have an x and a
y, xy, and then all of the

00:02:09.280 --> 00:02:12.900
constants, when you multiply
them, you get negative 60.

00:02:12.900 --> 00:02:16.150
And then finally, you have a
plus negative 5y squared.

00:02:16.150 --> 00:02:20.980
So what is negative 5 squared?

00:02:20.980 --> 00:02:25.930
Negative 5 squared
is positive 25.

00:02:25.930 --> 00:02:29.870
And then y squared is
just y squared.

00:02:29.870 --> 00:02:31.120
And we're done!

00:02:31.120 --> 00:02:36.420
We found the area of the square
with side 6x minus 5y.

00:02:36.420 --> 00:02:40.710
It's represented by this
expression right here.


Example 2: Adding polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:05.710
We're asked to simplify 5x
squared plus 8x minus 3 plus

00:00:05.710 --> 00:00:10.390
2x squared minus 7x plus 13x.

00:00:10.390 --> 00:00:12.830
So really, all we have to do
is we have to combine like

00:00:12.830 --> 00:00:16.490
terms-- terms that have x raised
to the same power.

00:00:16.490 --> 00:00:19.140
And the first thing we can do,
we can actually get rid of

00:00:19.140 --> 00:00:21.170
these parentheses right here,
because we have this whole

00:00:21.170 --> 00:00:23.470
expression, and then we're
adding it to this whole

00:00:23.470 --> 00:00:23.980
expression.

00:00:23.980 --> 00:00:26.890
The parentheses really don't
change our order

00:00:26.890 --> 00:00:28.230
of operations here.

00:00:28.230 --> 00:00:30.870
So let me just rewrite it once
without the parentheses.

00:00:30.870 --> 00:00:41.410
So we have 5x squared plus 8x
minus 3 plus 2x squared.

00:00:41.410 --> 00:00:43.190
If this was a minus then we'd
have to distribute the

00:00:43.190 --> 00:00:44.790
negative sign, but it's not.

00:00:44.790 --> 00:00:49.775
So plus 2x squared minus
7x plus 13x.

00:00:49.775 --> 00:00:52.670
Now let's just look at the
different terms that have

00:00:52.670 --> 00:00:53.850
different degrees of x.

00:00:53.850 --> 00:00:57.170
Let's start with the x squared
terms. So you have a 5x

00:00:57.170 --> 00:01:00.570
squared term here and you have a
2x squared term right there.

00:01:00.570 --> 00:01:04.235
So 5 of something plus 2 of that
same something is going

00:01:04.235 --> 00:01:05.480
to be 7 of that something.

00:01:05.480 --> 00:01:09.080
So that's going to
be 7x squared.

00:01:09.080 --> 00:01:11.420
And then let's look at
the x terms here.

00:01:11.420 --> 00:01:13.970
So we have an 8x right there.

00:01:13.970 --> 00:01:16.300
We have a minus 7x.

00:01:16.300 --> 00:01:18.900
And then we have a plus 13x.

00:01:18.900 --> 00:01:21.240
So if you have 8 of something
minus 7 of something, you're

00:01:21.240 --> 00:01:22.700
just going to have 1
of that something.

00:01:22.700 --> 00:01:25.400
And then if you add 14 of
that something more,

00:01:25.400 --> 00:01:26.330
you're going to 15.

00:01:26.330 --> 00:01:29.910
So this is going
to be plus 15x.

00:01:29.910 --> 00:01:31.980
8x minus 7x-- oh, sorry.

00:01:31.980 --> 00:01:33.310
You're going to have 14x.

00:01:33.310 --> 00:01:36.800
8 minus 7 is 1 plus 13 is 14.

00:01:36.800 --> 00:01:38.655
Plus 14x.

00:01:38.655 --> 00:01:42.280
That's these three terms.
8x minus 7x plus 13x.

00:01:42.280 --> 00:01:45.550
And then finally, you have a
negative 3-- or minus 3,

00:01:45.550 --> 00:01:46.580
depending on how you
want to view it.

00:01:46.580 --> 00:01:48.600
And that's the only
constant term.

00:01:48.600 --> 00:01:50.600
You could say it's x
times x to the 0.

00:01:50.600 --> 00:01:52.320
But it's a constant term.

00:01:52.320 --> 00:01:53.690
It's not be multiplied by x.

00:01:53.690 --> 00:01:56.590
And that's the only one
there, so minus 3.

00:01:56.590 --> 00:01:59.070
And we've simplified it
as far as we can go.

00:01:59.070 --> 00:02:00.750
We are done.


Polynomials 1

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:03.990
Write a binomial to express
the difference between the

00:00:03.990 --> 00:00:08.109
area of a rectangle with length
p and width 2r and the

00:00:08.109 --> 00:00:10.440
area of a circle with
diameter 4r.

00:00:10.440 --> 00:00:13.610
And they tell us that p
is greater than 4r.

00:00:13.610 --> 00:00:15.640
So the first thing that might
be a little confusing, say,

00:00:15.640 --> 00:00:16.600
write a binomial.

00:00:16.600 --> 00:00:17.750
You're like, what's
a binomial?

00:00:17.750 --> 00:00:20.930
All a binomial is is an
expression with two terms. And

00:00:20.930 --> 00:00:22.560
really, just to do this,
you pretty much

00:00:22.560 --> 00:00:23.720
have to write a binomial.

00:00:23.720 --> 00:00:25.470
I'll show you what
that's all about.

00:00:25.470 --> 00:00:30.400
So they want us to express the
difference between-- and I'll

00:00:30.400 --> 00:00:31.220
color code this.

00:00:31.220 --> 00:00:37.060
Between-- in pink-- the area of
a rectangle with length p

00:00:37.060 --> 00:00:45.590
and width 2r, and the area of
a circle with diameter 4r.

00:00:45.590 --> 00:00:47.350
So let's think about each
of these independently.

00:00:47.350 --> 00:00:50.710
So we have the area of a
rectangle first. So the area

00:00:50.710 --> 00:00:55.450
of a rectangle, let's say
that is our rectangle.

00:00:55.450 --> 00:00:58.300
And it has a length
p and width 2r.

00:01:01.770 --> 00:01:04.830
So we know the area of a
rectangle is going to be the

00:01:04.830 --> 00:01:06.530
length times the width.

00:01:06.530 --> 00:01:10.800
So the area here is going
to be p times 2r.

00:01:10.800 --> 00:01:13.220
We could say 2rp.

00:01:13.220 --> 00:01:14.920
That is the area of
the rectangle.

00:01:14.920 --> 00:01:18.220
Let's think about the area of
the circle with diameter 4r.

00:01:18.220 --> 00:01:21.460
So let me draw a circle here.

00:01:21.460 --> 00:01:22.840
That's our circle.

00:01:22.840 --> 00:01:27.180
The diameter of the
circle is 4r.

00:01:27.180 --> 00:01:32.050
And we know that the area
of a circle is 4pi

00:01:32.050 --> 00:01:34.440
times the radius squared.

00:01:34.440 --> 00:01:36.130
Oh actually, I almost
have made a mistake.

00:01:36.130 --> 00:01:38.390
The diameter is 4r,
not the radius.

00:01:38.390 --> 00:01:39.650
Let me be very clear.

00:01:39.650 --> 00:01:40.880
Diameter 4r.

00:01:40.880 --> 00:01:43.000
So diameter is two radiuses.

00:01:43.000 --> 00:01:45.970
This whole thing, this whole
distance-- I almost made a

00:01:45.970 --> 00:01:48.860
blunder-- this whole
distance is 4r.

00:01:48.860 --> 00:01:50.780
The diameter is 4r.

00:01:50.780 --> 00:01:52.970
So the radius is going
to be 1/2 that.

00:01:52.970 --> 00:01:54.560
The radius is going to be 2r.

00:01:58.060 --> 00:02:01.830
And we know that the area
of a circle is pi

00:02:01.830 --> 00:02:03.720
times the radius squared.

00:02:03.720 --> 00:02:06.110
In this case, the
radius is 2r.

00:02:06.110 --> 00:02:13.820
pi times 2r squared, which is
equal to pi times 4r squared.

00:02:13.820 --> 00:02:15.840
2 squared times r squared.

00:02:15.840 --> 00:02:19.670
Or we could just say that's
4pi r squared.

00:02:19.670 --> 00:02:23.300
Now, they want us to write
a binomial to express the

00:02:23.300 --> 00:02:26.650
difference between this thing,
the difference between the

00:02:26.650 --> 00:02:30.650
area of the rectangle and
the area of the circle.

00:02:30.650 --> 00:02:32.790
And this thing right
over here.

00:02:32.790 --> 00:02:33.760
So what's the difference?

00:02:33.760 --> 00:02:36.050
Well, it's going to be this
thing minus that thing.

00:02:36.050 --> 00:02:42.120
So it's going to be 2rp minus--
since I underlined

00:02:42.120 --> 00:02:46.360
difference in orange, I'll put
the minus in orange as well--

00:02:46.360 --> 00:02:51.260
minus 4pi r squared.

00:02:51.260 --> 00:02:52.130
And we're done.

00:02:52.130 --> 00:02:54.230
This is a binomial.

00:02:54.230 --> 00:02:56.150
We have two terms here.

00:02:56.150 --> 00:02:57.070
That's why it's binomial.

00:02:57.070 --> 00:02:59.600
If it had three terms,
it'd be trinomial.

00:02:59.600 --> 00:03:01.470
If there was kind of an
arbitrary number of terms with

00:03:01.470 --> 00:03:03.580
more than one, you'd call
it a polynomial.

00:03:03.580 --> 00:03:06.750
If you only have one term here,
you call it a monomial.

00:03:06.750 --> 00:03:09.850
So we are done.


Fractional exponent expressions 2

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:04.256
We're asked to simplify r to the
2/3 s to the third, that

00:00:04.256 --> 00:00:05.370
whole thing squared.

00:00:05.370 --> 00:00:10.050
Times the square root of 20r to
the fourth s to the fifth.

00:00:10.050 --> 00:00:12.040
Now this looks kind of daunting,
but I think if we

00:00:12.040 --> 00:00:14.480
take it step by step it
shouldn't be too bad.

00:00:14.480 --> 00:00:17.100
So first we can look at this
first expression right here

00:00:17.100 --> 00:00:19.710
where we're taking this product
to the second power.

00:00:19.710 --> 00:00:23.390
We know that instead we can take
each of the terms in the

00:00:23.390 --> 00:00:25.410
product to the second power
and then take the product.

00:00:25.410 --> 00:00:30.240
So this is going to be the same
thing as r to the 2/3

00:00:30.240 --> 00:00:36.560
squared times s to the
third squared.

00:00:36.560 --> 00:00:38.560
And now let's look at this
radical over here.

00:00:38.560 --> 00:00:40.690
We have the square root, but
that's the exact same thing as

00:00:40.690 --> 00:00:44.030
raising something to
the 1/2 power.

00:00:44.030 --> 00:00:47.340
So this is equal to--
so times this part.

00:00:47.340 --> 00:00:49.330
Let me do this in a
different color.

00:00:49.330 --> 00:00:54.090
This part right here, that
is the same thing as 20.

00:00:54.090 --> 00:00:56.540
And instead of just writing
20, let me write 20 as the

00:00:56.540 --> 00:00:59.210
product of a perfect square
and a non-perfect square.

00:00:59.210 --> 00:01:04.230
So 20 is the same thing
as 4 times 5.

00:01:04.230 --> 00:01:05.990
That's the 20 part.

00:01:05.990 --> 00:01:10.250
Times r to the fourth times
s to the fifth.

00:01:10.250 --> 00:01:12.750
Now let me write s to the fifth
also as a product of a

00:01:12.750 --> 00:01:14.510
perfect square and a
non-perfect square.

00:01:14.510 --> 00:01:16.440
r to the fourth is obviously
a perfect square.

00:01:16.440 --> 00:01:18.080
Its square root is r squared.

00:01:18.080 --> 00:01:19.840
But let's write s to the
fifth in a similar way.

00:01:19.840 --> 00:01:22.830
So s to the fifth we can
rewrite as s to the

00:01:22.830 --> 00:01:25.150
fourth times s.

00:01:25.150 --> 00:01:25.280
Right?

00:01:25.280 --> 00:01:29.010
S to the fourth times s to the
first, that is s to the fifth.

00:01:29.010 --> 00:01:33.480
And of course, all of this has
to be raised to the 1/2 power.

00:01:33.480 --> 00:01:35.720
Now let's simplify
this even more.

00:01:35.720 --> 00:01:38.790
If we're taking something to the
2/3 power and then to the

00:01:38.790 --> 00:01:41.290
second power, we can just
multiply the exponents.

00:01:41.290 --> 00:01:45.030
So this term right here,
we can simplify this as

00:01:45.030 --> 00:01:47.420
r to the 4/3 power.

00:01:47.420 --> 00:01:49.560
And just as a bit of review,
taking something to the 4/3

00:01:49.560 --> 00:01:52.910
power, you can view it as either
taking-- finding its

00:01:52.910 --> 00:01:55.790
cube root, taking it to the 1/3
power, and then taking its

00:01:55.790 --> 00:01:57.470
cube root to the fourth power.

00:01:57.470 --> 00:02:00.130
Or you can view it as taking
it to the fourth power and

00:02:00.130 --> 00:02:02.070
then finding the cube
root of that.

00:02:02.070 --> 00:02:06.090
Those are both legitimate ways
of something being raised to

00:02:06.090 --> 00:02:07.210
the 4/3 power.

00:02:07.210 --> 00:02:11.840
So you have r to the 4/3 times
s to the 3 times 2.

00:02:11.840 --> 00:02:15.990
Times s to the sixth power.

00:02:15.990 --> 00:02:18.130
And then we could raise each of
these terms right here to

00:02:18.130 --> 00:02:19.720
the 1/2 power.

00:02:19.720 --> 00:02:22.685
So times-- let me color
code it a little bit.

00:02:22.685 --> 00:02:23.630
And we actually wouldn't
need the

00:02:23.630 --> 00:02:24.750
parentheses once we do that.

00:02:24.750 --> 00:02:33.800
Times 4 to the 1/2 times
5 to the 1/2.

00:02:33.800 --> 00:02:35.520
That term right there.

00:02:35.520 --> 00:02:42.920
Times r to the fourth
to the 1/2 power.

00:02:42.920 --> 00:02:45.620
Times-- I might run out of
colors-- s to the fourth to

00:02:45.620 --> 00:02:46.870
the 1/2 power.

00:02:49.680 --> 00:02:52.670
We're raising each of these
terms to that 1/2 power.

00:02:52.670 --> 00:02:58.380
Times s to the 1/2 power.

00:02:58.380 --> 00:03:00.990
There's a lot of ways we can
go with this, but the one

00:03:00.990 --> 00:03:02.800
thing that might jump out is
that there are some perfect

00:03:02.800 --> 00:03:05.280
squares here and we're raising
them to the 1/2 power.

00:03:05.280 --> 00:03:06.170
We're taking their
square roots, so

00:03:06.170 --> 00:03:07.800
let's simplify those.

00:03:07.800 --> 00:03:10.890
So this 4 to the 1/2, that's
the same thing as 2.

00:03:10.890 --> 00:03:13.550
We're taking the principal
root of 4.

00:03:13.550 --> 00:03:14.770
5 to the 1/2?

00:03:14.770 --> 00:03:17.050
Well, we can't take the square
root of that, so let's just

00:03:17.050 --> 00:03:18.385
write that as the square
root of 5.

00:03:21.210 --> 00:03:24.780
r to the fourth to the 1/2.

00:03:24.780 --> 00:03:25.900
There's two ways you
can think about it.

00:03:25.900 --> 00:03:28.330
4 times 1/2 is 2.

00:03:28.330 --> 00:03:29.590
So this is r squared.

00:03:29.590 --> 00:03:31.140
Or you could say the
square root of r to

00:03:31.140 --> 00:03:32.800
the fourth is r squared.

00:03:32.800 --> 00:03:35.760
So this is r squared.

00:03:35.760 --> 00:03:40.270
Similarly, the square root of
s to the fourth or s to the

00:03:40.270 --> 00:03:42.780
1/2 is also s squared.

00:03:42.780 --> 00:03:44.900
And then this s to the 1/2,
let's just write that as the

00:03:44.900 --> 00:03:46.490
square root of s.

00:03:50.370 --> 00:03:52.380
Just like that.

00:03:52.380 --> 00:03:56.900
Let's see what else
we can do here.

00:03:56.900 --> 00:04:01.030
Let me write these other terms.
We have an r to the 4/3

00:04:01.030 --> 00:04:05.830
times s to the sixth times 2
times square root of 5 times r

00:04:05.830 --> 00:04:09.840
squared times s squared times
the square root of s.

00:04:09.840 --> 00:04:11.800
Now, a couple of things
we can do here.

00:04:11.800 --> 00:04:14.490
We could combine these s
terms. Let's do that.

00:04:14.490 --> 00:04:17.230
Actually, just write the 2 out
front first. So let's write

00:04:17.230 --> 00:04:20.910
the 2 out front first.
So you have 2 times.

00:04:20.910 --> 00:04:23.990
Now let's look at these
two s terms over here.

00:04:23.990 --> 00:04:26.770
We have s to the sixth
times s squared.

00:04:26.770 --> 00:04:28.500
When someone says to simplify
it, there's multiple

00:04:28.500 --> 00:04:29.560
interpretations for it.

00:04:29.560 --> 00:04:32.650
But we'll just say s to the
sixth times s squared.

00:04:32.650 --> 00:04:33.840
That's s to the eighth.

00:04:33.840 --> 00:04:34.760
6 plus 2.

00:04:34.760 --> 00:04:38.290
Times s to the eighth power.

00:04:38.290 --> 00:04:40.800
Times-- now this one's
interesting and we might want

00:04:40.800 --> 00:04:45.040
to break it up depending on what
we consider to be truly

00:04:45.040 --> 00:04:45.630
simplified.

00:04:45.630 --> 00:04:48.516
We have r to the 4/3
times r squared.

00:04:52.110 --> 00:04:55.440
r to the 4/3 is the same thing
as r to the 1 and 1/3.

00:04:55.440 --> 00:04:57.040
That's what 4/3 is.

00:04:57.040 --> 00:05:02.750
So 1 and 1/3 plus
2 is 3 and 1/3.

00:05:02.750 --> 00:05:06.520
So we could write this times
r to the 3 and 1/3.

00:05:06.520 --> 00:05:07.840
That's a little inconsistent.

00:05:07.840 --> 00:05:09.460
Over here I'm adding
a fraction.

00:05:09.460 --> 00:05:12.850
Over here with the s I kind of
left out the s to the 1/2 from

00:05:12.850 --> 00:05:13.545
the s's here.

00:05:13.545 --> 00:05:15.840
But we could play around with
it and all of those would be

00:05:15.840 --> 00:05:16.660
valid expressions.

00:05:16.660 --> 00:05:18.050
So we've already dealt
with the 2.

00:05:18.050 --> 00:05:19.960
We've already dealt with
these two s's.

00:05:19.960 --> 00:05:22.190
We've already dealt
with these r's.

00:05:22.190 --> 00:05:24.020
And then you have the square
root of 5 times the

00:05:24.020 --> 00:05:24.770
square root of s.

00:05:24.770 --> 00:05:26.070
And we could merge them
if we want, but I

00:05:26.070 --> 00:05:27.610
won't do it just yet.

00:05:27.610 --> 00:05:34.790
Times the square root of 5 times
the square root of s.

00:05:34.790 --> 00:05:36.170
Now there's two ways
we could do it.

00:05:36.170 --> 00:05:38.650
We might not like having a
fractional exponent here.

00:05:38.650 --> 00:05:39.840
And then we could
break it out.

00:05:39.840 --> 00:05:42.430
Or we might want to take this
guy and merge it with the

00:05:42.430 --> 00:05:43.410
eighth power.

00:05:43.410 --> 00:05:44.440
Because you know that
this is the same

00:05:44.440 --> 00:05:45.880
thing as s to the 1/2.

00:05:45.880 --> 00:05:48.300
So let's do it both ways.

00:05:48.300 --> 00:05:52.380
So if we wanted to merge all
of the exponents, we could

00:05:52.380 --> 00:05:57.900
write this as 2 times s to the
eighth times s to the 1/2.

00:05:57.900 --> 00:06:00.300
So s to the eighth
and s to the 1/2.

00:06:00.300 --> 00:06:03.430
That would be 2 times s
to the 8-- I can even

00:06:03.430 --> 00:06:04.410
write it as a decimal.

00:06:04.410 --> 00:06:05.480
8.5.

00:06:05.480 --> 00:06:09.850
8 plus-- you could imagine this
is s to the 0.5 power.

00:06:09.850 --> 00:06:14.630
So that's 8.5 times r
to the 3 and 1/3.

00:06:14.630 --> 00:06:16.260
I'm kind of mixing
notations here.

00:06:16.260 --> 00:06:17.740
I have just a decimal notation,
then I have a

00:06:17.740 --> 00:06:19.910
fraction notation, mixed
number notation.

00:06:19.910 --> 00:06:23.350
Times the square root of 5.

00:06:23.350 --> 00:06:24.700
This is one simplification.

00:06:24.700 --> 00:06:27.140
I kind of have it in the
fewest terms possible.

00:06:27.140 --> 00:06:29.420
The other simplification if you
don't want to have these

00:06:29.420 --> 00:06:33.170
fractional exponents out here,
you could write it as-- I'll

00:06:33.170 --> 00:06:34.840
do this in a different color.

00:06:34.840 --> 00:06:36.790
You could write this-- and
these are all equivalent

00:06:36.790 --> 00:06:37.330
statements.

00:06:37.330 --> 00:06:40.050
So it's up to debate what
simplified really means.

00:06:40.050 --> 00:06:45.270
So you could write this as
2 times s to the eighth.

00:06:45.270 --> 00:06:47.950
Instead of writing r to the 3
and 1/3, we could write r to

00:06:47.950 --> 00:06:53.870
the third times the cube root of
r, which is the same thing

00:06:53.870 --> 00:06:54.720
as r to the 1/3.

00:06:54.720 --> 00:06:57.250
We could write r to the third
times r to the 1/3.

00:06:57.250 --> 00:07:01.000
r to the 1/3 is the same thing
as the cube root of r.

00:07:01.000 --> 00:07:02.730
And then you have the square
root of these two guys.

00:07:02.730 --> 00:07:05.390
Both of these guys are being
raised to the 1/2 power.

00:07:05.390 --> 00:07:09.830
So you could then say times
the square root of 5s.

00:07:09.830 --> 00:07:12.300
I like this one a little bit
more, the one on the left.

00:07:12.300 --> 00:07:14.070
To me this is really
simplified.

00:07:14.070 --> 00:07:18.030
We've merged all of the bases.

00:07:18.030 --> 00:07:20.190
We have these two numbers, we've
merged all the s terms,

00:07:20.190 --> 00:07:22.370
all the r terms. This is a
little bit more complicated.

00:07:22.370 --> 00:07:23.200
You have a cube root.

00:07:23.200 --> 00:07:25.130
You haven't separated
the s's and the r's.

00:07:25.130 --> 00:07:27.870
So I would go with this one if
someone really wanted me--

00:07:27.870 --> 00:07:30.680
said hey, Sal, simplify
it how you like.


Multiplying and dividing monomials 3

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.740
Simplify the expression 28a to
the fourth, b to the sixth,

00:00:04.740 --> 00:00:08.370
over 7a to the ninth b.

00:00:08.370 --> 00:00:11.380
So a good place to start here
is to just try to kind of

00:00:11.380 --> 00:00:14.520
break this up into things that
we can simplify a lot easier.

00:00:14.520 --> 00:00:18.255
So the 28 and 7, we know that
28 is divisible by 7.

00:00:18.255 --> 00:00:22.490
So let's say we break that apart
into 28 times-- 28 over

00:00:22.490 --> 00:00:27.030
7-- times-- and then we can
break this one apart-- a to

00:00:27.030 --> 00:00:31.380
the fourth over a
to the ninth.

00:00:31.380 --> 00:00:35.100
So times a to the fourth
over a to the ninth.

00:00:35.100 --> 00:00:37.470
I'm just rewriting what we did
here, but I'm writing it as a

00:00:37.470 --> 00:00:41.020
product of three terms instead
of just one term over here.

00:00:41.020 --> 00:00:43.710
And then finally, you have this
last term over here, b to

00:00:43.710 --> 00:00:44.910
the sixth over b.

00:00:44.910 --> 00:00:48.680
So times b to the
sixth over b.

00:00:48.680 --> 00:00:51.430
And if you multiplied all this
out, you'd take the product of

00:00:51.430 --> 00:00:53.170
the numerators, you'd
get that up there.

00:00:53.170 --> 00:00:55.200
You get the product of the
denominators, you'd get this

00:00:55.200 --> 00:00:55.970
right there.

00:00:55.970 --> 00:00:58.130
The whole reason I did that is
because each of these terms

00:00:58.130 --> 00:01:00.170
can be simplified
pretty easily.

00:01:00.170 --> 00:01:02.290
This 28 over 7, that's easy.

00:01:02.290 --> 00:01:07.550
28 divided by 7 is 4.

00:01:07.550 --> 00:01:09.770
Then we have the
a to the fourth

00:01:09.770 --> 00:01:11.790
divided by a to the ninth.

00:01:11.790 --> 00:01:14.240
And when you have an exponent
in the denominator, you can

00:01:14.240 --> 00:01:15.850
subtract it from the exponent
in the numerator.

00:01:15.850 --> 00:01:17.930
We have the same base,
a, that's why I

00:01:17.930 --> 00:01:19.020
grouped it this way.

00:01:19.020 --> 00:01:24.080
So this is going to be equal to
a to the 4 minus 9 power.

00:01:24.080 --> 00:01:25.870
What's 4 minus 9?

00:01:25.870 --> 00:01:27.510
It's negative 5.

00:01:27.510 --> 00:01:29.110
4a to the negative 5.

00:01:29.110 --> 00:01:30.890
4 minus 9.

00:01:30.890 --> 00:01:33.080
And the same thing here
with the b's.

00:01:33.080 --> 00:01:34.280
Times b.

00:01:34.280 --> 00:01:36.080
6 minus-- you have a 1 here.

00:01:36.080 --> 00:01:37.570
You don't see the 1,
but it's there.

00:01:37.570 --> 00:01:38.850
If you just have a b, that's
the same thing

00:01:38.850 --> 00:01:40.420
as b to the 1 power.

00:01:40.420 --> 00:01:44.360
So b to the sixth divided by b
to the 1 power is b to the

00:01:44.360 --> 00:01:47.240
sixth minus 1, or b to
the fifth power.

00:01:47.240 --> 00:01:48.060
And we're done.

00:01:48.060 --> 00:01:49.840
We've simplified
the expression.

00:01:49.840 --> 00:01:52.750
If you don't want a negative
exponent there, you could say,

00:01:52.750 --> 00:01:59.450
well, this is the same thing
as 4 times 1 over a to the

00:01:59.450 --> 00:02:03.190
fifth-- that's the same thing as
a to the negative 5; these

00:02:03.190 --> 00:02:07.740
two things are the exact same
thing-- times b to the fifth.

00:02:07.740 --> 00:02:11.900
And then this, of course, would
be equal to 4 b to the

00:02:11.900 --> 00:02:14.500
fifth, a to the fifth.

00:02:14.500 --> 00:02:16.660
Or you could put 4 b to the
fifth over a to the fifth.

00:02:16.660 --> 00:02:18.850
All of these are legitimate
ways, but all of these are

00:02:18.850 --> 00:02:22.350
simplified versions of our
original expression.


Example 2: Multiplying two monomials with two variables

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:04.940
Express the area of a rectangle
with length 4xy and

00:00:04.940 --> 00:00:07.580
width 2y as a monomial.

00:00:07.580 --> 00:00:11.360
Monomial just means
just one term.

00:00:11.360 --> 00:00:13.710
So let's think about
a rectangle.

00:00:13.710 --> 00:00:15.910
So let me draw a
rectangle here.

00:00:19.350 --> 00:00:24.860
And they're telling us that the
length is 4xy and they're

00:00:24.860 --> 00:00:28.850
telling us that the
width is 2y.

00:00:28.850 --> 00:00:31.610
And just as a bit of a
refresher, we know the area of

00:00:31.610 --> 00:00:35.950
a rectangle is just the width
times the length, or the

00:00:35.950 --> 00:00:38.045
height times the width, or
however you want to view it--

00:00:38.045 --> 00:00:40.680
just the product of its
two dimensions.

00:00:40.680 --> 00:00:45.390
So the area of this rectangle,
the area is going to be equal

00:00:45.390 --> 00:00:58.390
to this length, 4xy times
the width times 2y.

00:00:58.390 --> 00:00:59.860
We can simplify this.

00:00:59.860 --> 00:01:01.540
We have a 4 times a 2.

00:01:05.570 --> 00:01:07.660
Whenever you're just taking a
product of a bunch of things

00:01:07.660 --> 00:01:09.680
you can switch the order however
you like, as long as

00:01:09.680 --> 00:01:11.500
it's all multiplication.

00:01:11.500 --> 00:01:14.620
So 4 times 2, that gives us 8.

00:01:14.620 --> 00:01:16.590
Then we have this
x sitting here.

00:01:16.590 --> 00:01:20.380
That is the only x we have
there, so it's 8 times x.

00:01:20.380 --> 00:01:23.310
And then we have a y here.

00:01:23.310 --> 00:01:25.260
We could view that as y to the
first. And then we have

00:01:25.260 --> 00:01:26.220
another y there.

00:01:26.220 --> 00:01:29.550
We could view that as y to
the first. So y times y.

00:01:29.550 --> 00:01:31.140
You could view that
as y squared.

00:01:31.140 --> 00:01:33.160
Or you could say look, y to
the first times y to the

00:01:33.160 --> 00:01:37.140
first-- same base, add the two
exponents-- 1 plus 1 is 2.

00:01:37.140 --> 00:01:40.970
So it's 8xy squared.

00:01:40.970 --> 00:01:45.670
This y squared covers both that
y and that y over there.

00:01:45.670 --> 00:01:46.230
And we're done.

00:01:46.230 --> 00:01:49.855
We've expressed the area of this
rectangle as a monomial.


Applying radical equations 1

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:04.770
The distance a person can see on
a clear day is estimated by

00:00:04.770 --> 00:00:10.930
the formula v is equal to 1,225
times the square root of

00:00:10.930 --> 00:00:14.760
a, all of that over 1,000,
where v is visibility.

00:00:14.760 --> 00:00:18.540
So that right there is
visibility in miles, and a is

00:00:18.540 --> 00:00:20.460
equal to altitude.

00:00:20.460 --> 00:00:21.750
Fair enough.

00:00:21.750 --> 00:00:28.770
Now, they say if Jerome can see
for 150 miles-- so that's

00:00:28.770 --> 00:00:32.960
how far he could see; so that is
the visibility, that is v--

00:00:32.960 --> 00:00:35.680
about how far above
the ground is he?

00:00:35.680 --> 00:00:38.250
So they want to know what
the altitude is.

00:00:38.250 --> 00:00:41.250
About how far off the
ground is he?

00:00:41.250 --> 00:00:42.130
This is a.

00:00:42.130 --> 00:00:43.540
And this is what
we don't know.

00:00:43.540 --> 00:00:46.040
We know that his visibility
is 150 miles.

00:00:46.040 --> 00:00:47.800
We want to know what a is.

00:00:47.800 --> 00:00:50.680
So we could look at this formula
for visibility as a

00:00:50.680 --> 00:00:52.310
function of altitude here.

00:00:52.310 --> 00:00:53.770
We know what the
visibility is.

00:00:53.770 --> 00:00:55.580
It is 150 miles.

00:00:55.580 --> 00:00:57.830
v is 150 miles.

00:00:57.830 --> 00:01:08.560
So we can say that v, or 150
miles, is equal to 1,225 times

00:01:08.560 --> 00:01:12.790
the square root of a-- a is how
far above the ground he

00:01:12.790 --> 00:01:21.860
is-- times the square root of
a, all of that over 1,000.

00:01:21.860 --> 00:01:24.420
Now we can multiply both
sides of this equation.

00:01:24.420 --> 00:01:26.700
We want to isolate this radical
here, so that we can

00:01:26.700 --> 00:01:27.680
solve for it.

00:01:27.680 --> 00:01:29.910
What we can do is we can
multiply both sides of this

00:01:29.910 --> 00:01:39.140
equation times 1,000
over 1,225.

00:01:39.140 --> 00:01:41.200
Let me scroll to the
left a little bit.

00:01:41.200 --> 00:01:45.130
1,000 over 1,225.

00:01:45.130 --> 00:01:47.170
And the whole point of that is
so that we can isolate the

00:01:47.170 --> 00:01:48.120
square root of a.

00:01:48.120 --> 00:01:50.760
This 1,000 cancels
with that 1,000.

00:01:50.760 --> 00:01:54.280
That 1,225 cancels
with that 1,225.

00:01:54.280 --> 00:01:57.080
And we get-- and I always like
to have the variable I'm

00:01:57.080 --> 00:02:00.220
solving for on the left-hand
side, so let's put it there.

00:02:00.220 --> 00:02:02.150
So let me just switch
sides as well.

00:02:02.150 --> 00:02:07.540
So we get the square root of
a is equal to all of this

00:02:07.540 --> 00:02:17.270
business over here, is equal to
1,000 times this 150, all

00:02:17.270 --> 00:02:24.050
of that over 1,225.

00:02:24.050 --> 00:02:26.400
And it looks like there's
some simplification

00:02:26.400 --> 00:02:27.610
that we can do here.

00:02:27.610 --> 00:02:32.010
Both of these numbers seem
divisible by 25.

00:02:32.010 --> 00:02:35.850
This number up here, 150,
that's 25 times 6.

00:02:35.850 --> 00:02:36.880
So let me write this.

00:02:36.880 --> 00:02:40.280
This is 25 times 6.

00:02:40.280 --> 00:02:43.030
And this number right
here, what is this?

00:02:43.030 --> 00:02:51.740
This is 25 times-- let's
see, 12 times 4 is 48.

00:02:51.740 --> 00:02:53.190
And you're going to
have one more.

00:02:53.190 --> 00:02:56.740
25 times 49, if I'm doing
my math right.

00:02:56.740 --> 00:02:58.130
Let me double check that.

00:02:58.130 --> 00:03:01.130
49 times 25.

00:03:01.130 --> 00:03:04.300
9 times 5 is 45.

00:03:04.300 --> 00:03:08.280
4 times 5 is 20, plus 4 is 24.

00:03:08.280 --> 00:03:10.340
Get rid of that, put a 0 here.

00:03:10.340 --> 00:03:12.710
2 times 9 is 18.

00:03:12.710 --> 00:03:15.805
2 times 4 is 8, plus 1 is 9.

00:03:15.805 --> 00:03:17.280
We get a 5.

00:03:17.280 --> 00:03:19.140
Get a 12.

00:03:19.140 --> 00:03:20.640
Yep, it's 1,225.

00:03:20.640 --> 00:03:22.820
So this is 25 times 49.

00:03:22.820 --> 00:03:24.450
This is 25 times 6.

00:03:24.450 --> 00:03:25.770
Let me cancel this.

00:03:25.770 --> 00:03:26.350
Cross that out.

00:03:26.350 --> 00:03:29.350
This 25 cancels with that 25.

00:03:29.350 --> 00:03:32.540
And so, does it look-- you
know, 49 is 7 times 7.

00:03:32.540 --> 00:03:35.380
So that looks about as simple
as we can get it.

00:03:35.380 --> 00:03:41.210
So this is equal to-- we could
write it as 6,000 over 49 is

00:03:41.210 --> 00:03:44.870
equal to the square root of a.

00:03:44.870 --> 00:03:47.810
So if we want to solve for a,
we can square both sides of

00:03:47.810 --> 00:03:48.870
this equation.

00:03:48.870 --> 00:03:51.260
So let's do that.

00:03:51.260 --> 00:03:52.090
Let me rewrite this.

00:03:52.090 --> 00:03:58.300
So you have the square root of
a is equal to 6,000 over 49.

00:03:58.300 --> 00:04:01.890
We want to square both sides
of this equation.

00:04:01.890 --> 00:04:07.770
So we square both sides of this
equation and we will get

00:04:07.770 --> 00:04:12.960
a, the altitude, that how far
above the ground Jerome is, is

00:04:12.960 --> 00:04:20.970
going to be equal to-- well,
6,000 squared over 49 squared.

00:04:20.970 --> 00:04:24.542
Or you could view it as 6,000
divided by 49 squared.

00:04:24.542 --> 00:04:27.530
So let's figure out what
that number is just to

00:04:27.530 --> 00:04:28.540
get a sense of things.

00:04:28.540 --> 00:04:30.862
And let me get a calculator
out for this one.

00:04:30.862 --> 00:04:34.890
I'll just get a calculator
out.

00:04:34.890 --> 00:04:37.450
Let me clear it out.

00:04:37.450 --> 00:04:46.570
All right, so we want to do
6,000 squared divided by 49

00:04:46.570 --> 00:04:54.000
squared, which is equal to
1,400-- or, we'll round up,

00:04:54.000 --> 00:04:59.350
14,994 feet.

00:04:59.350 --> 00:05:07.070
This is equal to 14,993 feet.

00:05:07.070 --> 00:05:10.190
I'm assuming that this equation
is taking our feet,

00:05:10.190 --> 00:05:11.550
our altitude in feet.

00:05:11.550 --> 00:05:15.380
They didn't mention it here, but
I'm assuming that in feet,

00:05:15.380 --> 00:05:16.490
because that makes sense.

00:05:16.490 --> 00:05:19.510
If you can see for about 150
miles, it makes sense that

00:05:19.510 --> 00:05:23.220
you're about 15,000
feet in the air.