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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
How Many Heads of Garlic?
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.
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Discussion
Product of Powers Property
x2⋅x3⇓x2⋅x3=x2+3=x5
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.
x2y3(xy2)
CommutativePropMult
Commutative Property of Multiplication
x2y3(y2x)
AssociativePropMult
Associative Property of Multiplication
x2(y3y2)x
MultPow
am⋅an=am+n
x2(y5)(x)
CommutativePropMult
Commutative Property of Multiplication
x2(x)(y5)
MultPow
am⋅an=am+n
x3(y5)
Multiply
Multiply
x3y5
Example
Making Hand-Pulled Noodles
Ignacio is helping Ms. Ley make hand-pulled noodles.
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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Hint
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?
Solution
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=22
Ms. Ley told Ignacio to fold and stretch the noodles four more times.
2×2×2×2=24
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.
22×24
The Product of Powers Property can be used to add the powers.
22×24=26
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.
26
Multiply 2 by itself six times to evaluate this numerical expression.
26=2×2×2×2×2×2=64
This means that Ignacio made 64 noodles by folding and stretching the dough six times.
Example
Great Job, Ignacio!
To encourage Ignacio to keep doing a great job, Ms. Ley gives him a bonus each time he gets a compliment from a customer.
x⋅x⋅x=x3
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.
x2⋅x4⋅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.
(x2y)⋅x4⋅(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"]}}
Hint
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.
Solution
x2⋅x4⋅x⇓x2⋅x4⋅x1
Use the Product of Powers Property to multiply the monomials by adding the exponents.
x2⋅x4⋅x1=x2+4+1=x7
This means that Ignacio will get a bonus of x7. 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.
(x2y)⋅x4⋅(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.
(x2y)⋅x4⋅(xy)=(x2⋅y⋅x4)⋅xy
Next, use the Commutative Property of Multiplication so the same variables are next to each other. In this case, use it to write x4 next to x2. This will show the application of the Product of Powers Property more clearly.
(x2⋅y⋅x4)⋅(xy)=(x2⋅x4⋅y)⋅xy=(x6⋅y)⋅xy
Finish the simplification by combining the rest of the variables in a similar fashion.
(x6⋅y)⋅xy
AssociativePropMult
Associative Property of Multiplication
x6⋅(y⋅x⋅y)
CommutativePropMult
Commutative Property of Multiplication
x6⋅(x⋅y⋅y)
ProdToPowTwoFac
a⋅a=a2
x6⋅(x⋅y2)
AssociativePropMult
Associative Property of Multiplication
(x6⋅x)⋅y2
MultPow
am⋅an=am+n
x7⋅y2
Multiply
Multiply
x7y2
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
Multiplying Monomials
Discussion
Quotient of Powers Property
Example
Be Careful with Customers
Everyone has bad days and today was no exception. In addition to the four compliments from the customers, Ignacio also got two complaints.
x2x4
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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.
x2y2x4y3
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"]}}
Hint
a How do you divide monomials?
b Begin by splitting the quotient into a product of fractions. Use the Quotient of Powers Property.
Solution
a Ignacio got a total of four compliments and two complaints.
x2x4
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. x2x4=x4−2=x2
This means that Ignacio will only get the bonus from two compliments. Be more careful the next time, Ignacio!
x2y2x4y3=x2x4⋅y2y3
Next, use the Quotient of Powers Property for each division.
x2x4⋅y2y3=x4−2⋅y3−2=x2⋅y1
Any number raised to the power of 1 is equal to itself.
x2⋅y1=x2⋅y
Finally, multiply the monomials.
x2⋅y=x2y
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
Dividing Monomials
Example
Making Dumplings
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.
x2
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.
m3d2
Today is March 10th. 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"]}}
Hint
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.
Solution
a Consider the given monomial.
x2
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.
m3d2
Substitute 3 for m and 10 for d to evaluate the monomial. Then, simplify the expression by multiplying the resulting values of m3 and d2.
This means that today's combination is 2700. Now Ignacio can start working on the secret sauce!
Pop Quiz
Evaluating Monomials in One Variable
Pop Quiz
Evaluating Monomials in Two Variables
Closure
Counting the Heads of Garlic
Ignacio was asked to peel all the heads of garlic in the crate.
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 x2y bags of garlic and each bag has y3 bulbs. Multiply these monomials to find the number of heads of garlic each crate.
x2y⋅y3=x2y4
There are xz many crates, so this monomial needs to be multiplied by xz.
x2y4⋅xz
AssociativePropMult
Associative Property of Multiplication
x2⋅(y4⋅x)⋅z
CommutativePropMult
Commutative Property of Multiplication
x2⋅(x⋅y4)⋅z
AssociativePropMult
Associative Property of Multiplication
(x2⋅x)⋅(y4⋅z)
MultPow
am⋅an=am+n
x3⋅y4⋅z
Multiply
Multiply
x3y4z