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How To Multiply & Divide Monomials
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Apr 3, 2025
Now that we're familiar with powers, exponents and monomials, we'll learn how to multiply and divide monomials. Chapters:
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multiplying and dividing monomials all
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right this is a section this is a
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section that again is easy just like
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what we did yesterday the p and exponent
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we did P and exponent we did prime
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numbers and composite numbers you guys
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great on that so I feel like we don't
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need to waste too much time on it so I
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want to move on to the next so now we're
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going to talk about multiplying and
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dividing monomial right we I give you a
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definition of what a monomial is
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yesterday and we're going to just
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quickly review that again so a monom is
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pretty much a combination of a number
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and a variable in terms of
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multiplication or it could just be a
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variable that's isolated on by itself
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like X is called a monomial right 10 x
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is also a monomial because it's 10 * X
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okay 3x2 is also a monomial because it's
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3 * X2 and then x^ 4 is also a monomial
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is X raed to 4 now when you have
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something like this X s - y s that is
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not a monomial because these two
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expressions are separated by a
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subtraction shine right subtraction when
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you have a subtraction addition it's not
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a monomial yes so it's a monom something
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joined by multiplication joined by
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multiplication right this will be called
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later on if you go to algebra one polom
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why there's two there's two polinomial
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this is called a polom this is a
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monomial okay monomial is a single like
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you said yesterday one yeah so if it was
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if there was no um sign would it just
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would it be monmal if you were just like
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this x² y Square yes that would be
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monoral right CU this the expression is
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just these two are linked by
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multiplication as soon as you add a
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addition subtraction is no longer called
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a mon okay
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do that's monom okay because need to
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connect it as long as they connected by
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multiplication yeah monom all right now
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so we're going to talk about how how do
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you multiply these expressions and how
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do you divide it if you have a hard time
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understanding this I suggest you take
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notes don't look at me I'm not a clown
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I'm not a comedian I'm teaching you
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something I need you to pay attention
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right so take notes if you don't
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understand it if folding your hands and
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looking they not help all right so you
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need to take notes so now we have the
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the the first thing that we going to
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learn is called the product of powers
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it's a rule that deals with monom so
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it's a rule so how do you apply that
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rule so and how does that rule appli so
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when you multiplying two monomial that
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have the same base okay so
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example if you look here right I put X
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ra to the power of m mtip x ra to the
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power of n if you pay attention to this
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these two expressions you have a
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question
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what um if it would it be a mon if if we
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had multiplication and division in
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between I'll get to
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that I'll get to that so let's stick to
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this for now right so here if you look
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here right what do you notice about this
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two expression I have X the M * X the N
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what is the first that you notice yeah
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they have the same base they have the
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same base this is key to this
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section must have the same base if they
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do not have the same base there is no
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way you can connect them the way I did
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it here right so the base must be
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identical yeah why is it addition if
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you're
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mul you're asking me to prove something
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that go ahead so it's it's just addition
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just a rule okay right it's just a rule
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that when you have two Pol two monomial
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of the same base when you're multiplying
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them you are technically adding their
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exponent why is it that way because look
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at it right if I go x 3 * x 2 right
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watch if I was to expand this what would
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that be time what and this will
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be how many X do I have five so it's X
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power of five this is why is an addition
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I right so this is why you add the
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exponent so that's a good question mhm
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so this is why you add the exponents
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because when you expand them you end up
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just adding them so to make life easy
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when the base is the same all you're
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doing is just add the exponent right
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example x 5 * x 3 if you were to go
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traditionally you would have expanded
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these term into x * x * x 5 * x * x * x
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three times and then that's going to be
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a lot to count so what we do is when the
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Bas is the same you just add the
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exponent so x 5 * x 3 gives you x 8 yes
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you first and then you next yeah okay so
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um for the exponent part there out there
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do we have to like when you write it out
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do you have to do that number plus the
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other
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one that what do you mean like this well
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that's what I did you to do it that plus
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and then just I mean if you can you can
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just put extra eight if you quick like
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that right
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yes for the for the how how come and
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not why would it be negative this is not
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a negative sign it's not this is just a
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I'm just saying this is part one part
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two and then part three okay right what
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was your question um so why do the
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exponents
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uh I guess Plus instead of
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multiply I just proved it right so let's
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say I have x 5 * x 3 right if I was to
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expand this what would I be x * X how
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many times four times all right and this
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will be x * X how many times three times
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right this is three so when you count it
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how much is
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that eight eight so this is why you add
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them because when you multiply you
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expanded that means you having more of
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this this is X to the 8 that make sense
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right so this is why just a rule that
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it's called the product of Fes we just
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add the exponent does it depend excuse
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me okay that's a w so you make it what
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you want to make it where a w right yeah
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does it depend on the base or the base
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has to be the same if the base is not
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identical you can't do this so is it
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like let's say 2x * 3x then this could
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happen well we get to that you ahead of
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me okay we'll get to that right we'll
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get to that but you ahead of me so we'll
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get to that so now I give you another
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simple one here right I
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B2 * B2 right same thing is B4 because
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you just add the exponent the base the
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same now if if you look down here now I
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have like a monomial again I have 2 x 3r
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* 4 X 5 now going back to your question
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what I did here so if you have 2X to 3
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power and then 4X to the 5th power right
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you want to rearrange it okay so it's
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going to be basically 2 * 4 * x 3 * x 5
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now I know that 2 * 4 is equal to 8
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right and then for the exponent I just
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going to add them so this is X to 8 okay
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so you can rearrange it or if you don't
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feel like you need to do this you can
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just go ahead and tell me this is 8 x 8
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some of you guys may not have that
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problem
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identifying the U the factors and the
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the the expression that I raised power
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whatever right so for some of you you
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like oh I know this is 8 x 8 I'm fine
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with it but if you want to be like
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traditional you can just go step by step
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and get the answer does that make sense
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okay so it all depends on how quick you
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understand the material if you feel like
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you don't need to expand it like this
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then go ahead and find it more power to
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you all right
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now
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here what do I have here now I have 5 a
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to the 2 *
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-38 2 and is is it
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x three behind X this yeah this is two
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yeah 2 x and three it's only X that's to
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the^ three and here it's only X to the^
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five so what you do is just you multiply
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them you just multiply the numbers and
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then for
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the X to the three and X to five you
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because the the Bas is same this is
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called the base the same you just add
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the exponent so that becomes 8 x 8
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that all right
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now let's say I have this here 5 a to 2
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* -38 and I want to put this I want to
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simplify this what do I
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do3 * 5 right be3 *
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5 a squ * 8 right a is a to the^ of what
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when there's nothing I mean it's not
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technically nothing it's a to the one
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right because a that means a to the 1
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when you have a is a ra power
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1 time what why would you um three 8 the
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power what three that's it right 2 + 1
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is three oh yeah plus I was right so
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this is this is3 * 5 that's 15 and this
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is a to the thir power right so the
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numbers and 3 that's -5 and then this
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one with the the same base with
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different exponent you just add the
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exponent okay any questions on
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that no questions all right so let's
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try let's try something a little bit um
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so I want you all to find this for
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me y 4 * Y 2 I also want you to find uh
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10 y 10
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x
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times -6
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x 4 tell me what this
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is we have y 4 * y^
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2 what you
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got does anybody agree this is y six yes
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I think it's y 8 am I wrong or right 2 *
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4 yeah trick you trick you know y 6 you
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don't multiply the EXP what do you do
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you add
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them yeah it's times when you add the
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exponent right CU y four we did it
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earlier right it's y * y am I really
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that right and then y s is what y * y so
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if you
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m so you multiply them that means you're
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adding you're doing this right 1 2 3 4 5
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six so the six Right add you're adding
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the exponent so
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basically you're adding the exponent
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you're
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mtip add exp why because when you expain
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them this is what you get you're adding
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because you keep add I know the one is I
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think all
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right the first one so let's do the
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first one and then we going to get to
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the second one right we have 10 I did
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the second one I want to know what you
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got in a minute uh it's probably wrong
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but I you have no confidence can I I got
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what you got I got it I
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got *
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X that's right that was right - 60 x 5
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that's right because you do a 10 * -6 is
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-60 and then X and then x 4 you add the
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X that's x 5 so be -60 x to the 5 so we
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added
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right the X has the one over so it's 1 +
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4 is 5 right and then 10 * -6 is -6 so
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it's 60 x 5 all right all right so now
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now that we understand this we're going
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to move on to division right
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[Music]
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so we going to talk about the division P
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of P rule so how does that apply I'm
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glad you ask I want someone to to see if
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you can figure it out right if I have x
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to the
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end x n what do you think that's going
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to give me X to
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what
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XS n m- n
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right but now we're going to prove it
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why is see like this right we're going
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to prove it so I'll give you an example
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and we going to prove it so if I have x
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5 over x 2 based on this rule it be x
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what 5 - 2 which is X 3 now let's prove
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it right so let's prove it so if I have
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x 5 over x 2 let me TR explain this what
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is that going to give me uh how many X's
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X
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1 2 3 4 5 what about they one
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under now when I divide what happens to
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these two here they can they can cancel
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out can they right so they can cancel
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out this can cancel out so you left with
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how many three three so this is y this
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is X this how came came up with this Ru
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here does that make sense right when
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you're dividing all you doing is
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subtracting the exponent you take the
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one up top you have to be careful how
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you do or it can't be so I've seen some
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people this you have x 3 x 4 x 2 and it
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go this is 2 - 3 no no no no it's always
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top minus bottom right always like that
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make sure you keep the order correctly
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it's m minus n does that make sense yes
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or no all right so this is how we do
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this one so we do a coup more work yeah
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go ahead
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when you go to uh algebra one or two
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you're going to learn how to deal with
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negative X actually no we going to we
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going to do in section 9.4 what am I
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talking about how dare you I thought I
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waset I thought I was yeah this our very
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next chapter is negative exponent
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so you want to kill us I want to do
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the stop
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I was writing that leave it alone I was
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writing that you erased it oh okay let
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me what is
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that
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cheeks
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I all right I love you I love you you
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change this all right so now let's some
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some examples all right examples so
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now the question is you're going to have
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to deal with like different type of
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problem right let me
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uh let me give you something here to so
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like so let's do this here oh my god so
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let's do
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a she should break dance she go the
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Olympics and break dance oh
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break imagine how that look all right so
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what would you get here 25 x 10 over
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5x do not erase this yet so what would I
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give
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you oh my
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God you have 25 x 10 over 5 x
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[Music]
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5 hey hey
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[Music]
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hey all right 5x and five right because
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divide 5 into 25 5 10 - 5 is 5 so 5 x 5
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all right well that was that's the
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answer yeah that's the answer so this
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one is pretty simple okay so now uh
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let's go up to like some more complex
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problem here so I'm going to mix them up
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right you you mean easy right can this
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just be like the unit we do for the rest
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of the because this is way easier yeah
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the complex I don't know how to put
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numbers on a graph what do you think I
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Amma
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I'm
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actually What's that
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name
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name the other one Ricky hello Ricky n
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Ricky you see more like a Ricky are you
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kidding sorry be quiet yes
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can if I
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just all right so let's say we have this
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so how would you do it yeah wait are
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those two two
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1.5 is
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1.5 that's definely so how would I what
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would you do here so you have
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yeahp what you say what what do you
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get how do you get positive three what
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do we have here we
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have1 it's one I was correct it's one
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I was
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correct what is
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it I'm only looking at what the base the
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base is the same so what is you give you
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what 1.5 to the power of what 10 10 you
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don't need to change anything the B is
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the same wa what do you do there the two
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numers but I thought division so I
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thought you minus it not add it
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I mean wrong side there's a
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multiplication what are you supposed to
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do when
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there's why are we leaving those the
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same I'm a ding oh my God the Bas is the
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same I did not think about that that's
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crazy the Bas is the same
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right what to do I get I get in a second
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now we have -4 get to it y to 6 right
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x y^ 2 * 2x 2 right so now4
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y1 I'm just
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kid yes what do we do so first you're
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going to put like terms together okay so
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I did4 * * 2 okay * y power 6 * y^ 2 *
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x^
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X what's going on all right you see that
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right this is the perfect breakdown you
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have to make
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sure expressions of the same like are
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together right so you have -4 * -10 * 2
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yes when you WR things like that with
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the exponents do you Mo exponent to low
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exponent it doesn't matter
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okay you can't do that but it doesn't it
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really doesn't matter like it doesn't
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change the
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the same thing with factor trees or
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factor trees you need to put it from
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like you my
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question about like what you supposed to
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do when it's X3 and Y when there's two
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of them what
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then science right now we have * 40 40 *
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2 that's 80 right so this whole thing
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gives you 80 so 80 plus 6 and 6 + 2 is8
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it's 80 Y8 * X5 X5 right Bas easy you
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just add X this is probably the easiest
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chapter that we're going to be dealing
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with this quarter all right