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Monomials are fundamental for the development of mathematical expressions. They are also closely related to multiplication and exponentiation. This lesson will discuss the multiplication and division of monomials.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Challenge

Ignacio got a part-time job at a restaurant. His first task is to peel some garlic. Ms. Ley, the restaurant owner, tells him to peel all the heads of garlic in a crate.

The crate contains eight bags of garlic. Each bag has nine heads of garlic. How many heads of garlic does Ignacio have to peel?{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"heads of garlic","answer":{"text":["72"]}}

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x","y","z"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"heads of garlic","answer":{"text":["x^3y^4z","xzy","y^4x^3z","y^4zx^3","zx^3y^4","zy^4x^3"]}}

Discussion

$x_{2}⋅x_{3}⇓x_{2}⋅x_{3} =x_{2+3}=x_{5} $

If a monomial has more than one variable, the Associative Property of Multiplication and the Commutative Property of Multiplication can be used to simplify the multiplication.
$x_{2}y_{3}(xy_{2})$

CommutativePropMult

Commutative Property of Multiplication

$x_{2}y_{3}(y_{2}x)$

AssociativePropMult

Associative Property of Multiplication

$x_{2}(y_{3}y_{2})x$

MultPow

$a_{m}⋅a_{n}=a_{m+n}$

$x_{2}(y_{5})(x)$

CommutativePropMult

Commutative Property of Multiplication

$x_{2}(x)(y_{5})$

MultPow

$a_{m}⋅a_{n}=a_{m+n}$

$x_{3}(y_{5})$

Multiply

Multiply

$x_{3}y_{5}$

Example

Ignacio is helping Ms. Ley make hand-pulled noodles.

They need to make the dough and then fold and stretch it several times. Some flour is added between each step to prevent the noodles from sticking. The number of noodles will double each time the dough strip is folded. Ignacio folds and stretches the dough two times.
Ms. Ley walks in and tells Ignacio that the noodles are still way too thick, so he should fold and stretch the dough four more times. When he is finished, Ignacio removes the ends of the folds so the noodles are separated.

a Write a numeric expression that represents how many noodles Ignacio ends up making after following Ms. Ley's instructions. Write the answer as a power of $2.$

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b Evaluate the numeric expression from Part A to find how many noodles Ignacio made.

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a Each time Ignacio folds the dough, he doubles the amount of noodles he is making.

b How do you find the power of a number?

a By folding and stretching the dough once, Ignacio gets two big and thick noodles. If he folds and stretches it again, he will now have **double** the previous amount. This means that he doubled the number of noodles $two$ times. This can be expressed as a numeric expression.

$2×2=2_{2} $

Ms. Ley told Ignacio to fold and stretch the noodles $four$ more times.
$2×2×2×2=2_{4} $

Notice that the power corresponds to the number of times Ignacio has stretched and folded, or doubled, the noodles. Since the entire amount of noodles is being doubled each time the dough is folded, the two monomials are multiplied.
$2_{2}×2_{4} $

The Product of Powers Property can be used to add the powers.
$2_{2}×2_{4}=2_{6} $

This numeric expression represents that Ignacio folding and stretching the noodles six times.
b To find how many noodles Ignacio made, start with the numeric expression written in Part A.

$2_{6} $

Multiply $2$ by itself $six$ times to evaluate this numerical expression.
$2_{6} =2×2×2×2×2×2=64 $

This means that Ignacio made $64$ noodles by folding and stretching the dough six times.
Example

To encourage Ignacio to keep doing a great job, Ms. Ley gives him a bonus each time he gets a compliment from a customer.

This bonus is a small percent increase of his normal total wage for the day. Ignacio does not know how much this percentage is, so he thinks of it as $x.$ This means that if he gets three compliments, his payment is multiplied by $x$ $three$ times.$x⋅x⋅x=x_{3} $

Ignacio gets two compliments from Table $1,$ four compliments from Table $2,$ and one compliment from Table $3.$ He writes an expression to show his compliment bonus for the day.
$x_{2}⋅x_{4}⋅x $

a What is the degree of the monomial?

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b Ms. Ley also gives Ignacio a different bonus for bussing a table in under two minutes. Ignacio thinks of this bonus as $y.$ He managed clean Tables $1$ and $3$ within two minutes, so he includes this in his total bonus.

$(x_{2}y)⋅x_{4}⋅(xy) $

What is the degree of the monomial for his total bonus for the day? {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["9"]}}

a How do you multiply monomials? Does it matter which table the compliment came from?

b Use the Associative Property of Multiplication and the Commutative Property of Multiplication.

$x_{2}⋅x_{4}⋅x⇓x_{2}⋅x_{4}⋅x_{1} $

Use the Product of Powers Property to multiply the monomials by adding the exponents.
$x_{2}⋅x_{4}⋅x_{1} =x_{2+4+1}=x_{7} $

This means that Ignacio will get a bonus of $x_{7}.$ The degree of this monomial is $7.$ Notice that this is also the total number of compliments Ignacio got from all the tables. This is because the percentage bonus is applied each time he receives the a compliment.
b Ignacio was able to clean Tables $1$ and $3$ quickly enough to get the extra bonus. Sadly, it took longer to bus Table $2$ and he missed getting the bonus for that table. At least he got four compliments from that table! Consider the revised expression showing the total bonus he has earned today.

$(x_{2}y)⋅x_{4}⋅(xy) $

The total degree of the monomial is most easily found when the expression is written in simplest form. To multiply monomials using the Product of Powers Property, the monomials need to have the same base. Begin by using the Associative Property of Multiplication to group the terms.
$(x_{2}y)⋅x_{4}⋅(xy)=(x_{2}⋅y⋅x_{4})⋅xy $

Next, use the Commutative Property of Multiplication so the same variables are next to each other. In this case, use it to write $x_{4}$ next to $x_{2}.$ This will show the application of the Product of Powers Property more clearly.
$(x_{2}⋅y⋅x_{4})⋅(xy) =(x_{2}⋅x_{4}⋅y)⋅xy=(x_{6}⋅y)⋅xy $

Finish the simplification by combining the rest of the variables in a similar fashion.
$(x_{6}⋅y)⋅xy$

AssociativePropMult

Associative Property of Multiplication

$x_{6}⋅(y⋅x⋅y)$

CommutativePropMult

Commutative Property of Multiplication

$x_{6}⋅(x⋅y⋅y)$

ProdToPowTwoFac

$a⋅a=a_{2}$

$x_{6}⋅(x⋅y_{2})$

AssociativePropMult

Associative Property of Multiplication

$(x_{6}⋅x)⋅y_{2}$

MultPow

$a_{m}⋅a_{n}=a_{m+n}$

$x_{7}⋅y_{2}$

Multiply

Multiply

$x_{7}y_{2}$

$7+2=9 $

This means the degree of the monomial is $9.$ This is also the total number of bonuses Ignacio got, no matter how he got them! Pop Quiz

Discussion

Example

Everyone has bad days and today was no exception. In addition to the four compliments from the customers, Ignacio also got two complaints.

Unfortunately, Ms. Ley is very strict and told Ignacio that for each complaint she receives about him, she would cancel out one of the compliment bonuses he gets. This situation can be represented with the following division of monomials.$x_{2}x_{4} $

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b The same applies if Ignacio forgets to clean a table! Besides getting four compliments and two complaints, Ignacio quickly bussed three tables, but forgot to clean two. Forgetting to clean a table cancels out the bonus from quickly cleaning a table.

$x_{2}y_{2}x_{4}y_{3} $

What is the degree of the monomial resulting from the above quotient? {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["3"]}}

a How do you divide monomials?

b Begin by splitting the quotient into a product of fractions. Use the Quotient of Powers Property.

a Ignacio got a total of $four$ compliments and $two$ complaints.

$x_{2}x_{4} $

The Quotient of Powers Property can be used to divide the monomials because the bases are the same. Do this by subtracting the exponent of the denominator from the exponent of the numerator. $x_{2}x_{4} =x_{4−2}=x_{2} $

This means that Ignacio will only get the bonus from two compliments. Be more careful the next time, Ignacio!
$x_{2}y_{2}x_{4}y_{3} =x_{2}x_{4} ⋅y_{2}y_{3} $

Next, use the Quotient of Powers Property for each division.
$x_{2}x_{4} ⋅y_{2}y_{3} =x_{4−2}⋅y_{3−2}=x_{2}⋅y_{1} $

Any number raised to the power of $1$ is equal to itself.
$x_{2}⋅y_{1}=x_{2}⋅y $

Finally, multiply the monomials.
$x_{2}⋅y=x_{2}y $

The degree of the resulting monomial is the sum of the exponents of its variable factors. $2+1=3 $

This means that the degree of the monomial is $3.$ Ignacio got a total of three bonuses. Not too bad, but Ignacio is determined to do better!
Pop Quiz

Example

It is time to make dumplings! Ignacio wants to help Ms. Ley with the dough. He plans to roll it out and cut it into small squares.

The following monomial shows how many squares Ignacio will have after cutting the dough sheet into an $x-$by-$x$ grid.$x_{2} $

a Evaluate the above monomial when $x=4$ to find how many dumplings can be made from the dough sheet.

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b While Ms. Ley stuffs and folds the dumplings, Ignacio is going to make the sauce. The recipe is stored in a safe with a combination that changes depending on the date.

$m_{3}d_{2} $

Today is March $10_{th}.$ This corresponds to $m=3$ and $d=10.$ Evaluate the monomial for the given values to find the combination to the safe. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["2700"]}}

a Substitute $4$ for $x$ and calculate the power.

b Substitute $3$ for $m$ and $10$ for $d,$ then calculate the powers. Multiply the resulting powers to find the total value of the monomial.

a Consider the given monomial.

$x_{2} $

Substitute $4$ for $x$ to evaluate the monomial.
This means that, by cutting the dough sheet in a $4$-by-$4$ grid, Ignacio will be able to make $16$ dumplings.
b This time the given monomial has two variables.

$m_{3}d_{2} $

Substitute $3$ for $m$ and $10$ for $d$ to evaluate the monomial. Then, simplify the expression by multiplying the resulting values of $m_{3}$ and $d_{2}.$
$m_{3}d_{2}$

SubstituteII

$m=3$, $d=10$

$3_{3}⋅10_{2}$

CalcPow

Calculate power

$27⋅100$

Multiply

Multiply

$2700$

Pop Quiz

Pop Quiz

Closure

Ignacio was asked to peel all the heads of garlic in the crate.

There are a total of $8$ bags of garlic, each containing $9$ bulbs. Multiply the number of bags by the number of heads of garlic in each bag.$8×9=72 $

This means that Ignacio has to peel $72$ heads of garlic. The same reasoning can be applied to monomials. Each crate has $x_{2}y$ bags of garlic and each bag has $y_{3}$ bulbs. Multiply these monomials to find the number of heads of garlic each crate.
$x_{2}y⋅y_{3}=x_{2}y_{4} $

There are $xz$ many crates, so this monomial needs to be multiplied by $xz.$
$x_{2}y_{4}⋅xz$

AssociativePropMult

Associative Property of Multiplication

$x_{2}⋅(y_{4}⋅x)⋅z$

CommutativePropMult

Commutative Property of Multiplication

$x_{2}⋅(x⋅y_{4})⋅z$

AssociativePropMult

Associative Property of Multiplication

$(x_{2}⋅x)⋅(y_{4}⋅z)$

MultPow

$a_{m}⋅a_{n}=a_{m+n}$

$x_{3}⋅y_{4}⋅z$

Multiply

Multiply

$x_{3}y_{4}z$