WEBVTT

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- [Voiceover] Alright, in this video,

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we're gonna be multiplying
monomials together.

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Let me give you an example of a monomial.

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4x squared, that's a monomial.

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Now, why?

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Well, mono means one, which
refers to the number of terms.

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So this 4x squared, this is all one term.

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So we're gonna be working
with things like that.

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What won't we be working with?

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Well what about 4x squared plus 5x.

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How many terms are there?

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4x squared's the first
term, 5x is the second term,

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so this is not a monomial,

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this is actually called a
binomial, because bi means two.

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Like your bicycle's got
two wheels, for example.

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So not yet, go on to the future videos

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if you're ready for binomials.

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But we're just gonna be working

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with multiplying monomials together.

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So can we grab an example to look at.

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By the end of this video,
it should be very easy

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for you to multiply this
monomial, 5x squared,

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by this monomial.

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And I'm actually just gonna
give you the answer right here.

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And then I'm gonna slowly
walk you through some other

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questions that will lead us to why.

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But the answer to this
is 20x to the eighth.

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20x to the eighth.

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Take a look at that, see if
you can notice a pattern.

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What did we do with the five
and the four to get the 20?

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What did we do with the two
and the six to get the eight?

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That's getting a little
ahead of ourselves though.

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Before we can dive in there,

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let's remember some of
the exponent properties.

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A very specific exponent property

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that you should've seen before.

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If we look at five squared
times five to the fourth power,

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what's that going to equal?

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Well, if you remember
your exponent property,

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we'll do a quick reminder
here, I always add my exponent.

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So five squared times
five to the fourth power

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is equal to five to the sixth power.

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What about three to the fourth power

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times three to the fifth power?

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Well, again, I always add my exponents.

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Four plus five is three
to the ninth power,

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and my base of three stays the same.

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Great, so if you remember that,

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now we're ready to really start

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multiplying monomials that are new to you.

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And the new thing there
is that we are going

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to have variables involved.

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So let's start, let's take a
look at two monomials here.

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The first monomial is 4x,
and the second one is just x.

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And the four, I don't have
another number to multiply by,

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just got the four.

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And can I simplify x times x?

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Well, that's equal to x squared.

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Remember if I just have a variable,

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and there's no exponent there,

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it's equivalent to having a one,

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so x to the first power
times x to the first power,

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I add my exponents like
we just talked about,

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and one plus one is equal to two.

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Great, so let's move
on to another one here.

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If I have 4t times 3t.

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Well, four times three
is gonna be equal to 12,

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so I've combined my coefficients.

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And then t times t, again,
think of a one being there,

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is going to be t squared.

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So the answer here is 12t squared.

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So let's keep going,

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and once you get into the rhythm of these,

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they become pretty alright.

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So what if I had 4p to
the fifth power times,

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let's say 5p to the third power.

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What would that equal?

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Well you're gonna notice a pattern here

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that we've been pickin' up on,

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which is that I'm always gonna
multiply my coefficients,

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so four times five, is going to equal 20.

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And I'm always going to add my exponents.

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So p to the fifth and p to the third

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is p to the eighth power.

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so I multiply four and five til we get 20,

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I add five and three to get eight.

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And if you really wanted
to see why that is,

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let's really dive in
here and let's break down

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this first term, let's
break down 4p to the fifth.

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I can write that out as four times p,

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times p, times p, times p,
times p, that's five of 'em.

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That's four and five p's.

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And then that second term I can write as

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times five times p, times p, times p.

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What I'm gonna do is I'm
gonna group my numbers,

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cause I can work with numbers together,

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so let's put four times
five at the very front,

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and then it just becomes a
matter of how many p's do I have?

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We'll put all of those together as well.

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So I had five p's, so
there's the first five,

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and then I had three more.

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And we can simplify this
crazy looking expression

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by just multiplying my four
and my five to be my 20,

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and then writing this with an exponent,

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that's the beauty of exponents,
that's why we have 'em,

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is we can write a crazy
expression like that

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as p to the eighth,

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and you'll notice that this is, of course,

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what we got the first time.

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So great.

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What about 5y to the sixth times

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negative 3y to the eighth power?

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Again, multiply the
coefficients, add the exponents,

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and I've got a simplified expression.

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Let's get really crazy here,
let's have a little fun.

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So we've noticed the pattern,
let's have a little fun.

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Just saying, I can, I can do more.

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Negative 9x to the fifth power times

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negative three, use parentheses there,

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when you have a negative in front,

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you always wanna use parentheses.

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Let's do x to the 107th power.

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If I would have showed you
this before this video,

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you would have said oh my goodness,

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there's nothing I can do, I'm boxed,

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there's no way out.

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But now you know that it's as
simple as follow the rules.

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We're going to multiply the coefficients,

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negative nine times negative three is 27.

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Two negatives is a positive
and nine times three is 27.

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I'm gonna add my powers.

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Five plus 107 is a hundred, ooh, not two,

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that was almost a mistake I made there.

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Let's get rid of that, give
me a second chance here.

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Life's all about second chances,

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five plus 107 is 112.

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And so, this crazy expression,

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which is two monomials, here's the first,

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here's the second, when
we multiply and simplify

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we get another monomial,
which is 27x to the 112th.

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I'm gonna leave you on a cliffhanger here.

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Which, I'm gonna show you a problem.

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What variable should we use?

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You notice I've been trying
to vary the variables up

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to show you that it just doesn't matter.

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That's an ugly five,
let's get rid of that.

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Give me a second chance with that one too.

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So let's look at 5x to the third power,

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times 4x to the sixth power.

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And I'm gonna show you a wrong answer.

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I had a student that asked to do this,

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and here's the wrong
answer that they gave me.

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They told me 9x to the 18th power.

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That's terribly wrong.

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What did they do wrong?

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What did they do wrong?

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I want you to think to yourself,

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what have we been talking about?

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What did they do with the five
and the four to get the nine?

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What should they have done?

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What did they do with the three
and the six to get the 18,

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and what should they have done?

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That's multiplying monomials by monomials.