8. Multiply and Divide Monomials
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Chapter 4
8.
Multiply and Divide Monomials
In this lesson, the focus is on understanding how to multiply and divide monomials, particularly using the Product of Powers Property. The lesson uses practical examples to demonstrate these mathematical operations. For instance, it discusses a scenario where Ignacio, working in a restaurant, has to peel garlic and make noodles. The mathematical principles are applied to these real-world tasks, such as calculating the number of noodles made after folding and stretching the dough multiple times. The Product of Powers Property is emphasized as a crucial tool for simplifying these calculations. The lesson also touches on the Quotient of Powers Property for division tasks. Overall, it aims to make the subject matter relatable and easier to grasp by applying it to everyday situations.
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Lesson Settings & Tools
| | 13 Theory slides |
| | 11 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Multiply and Divide Monomials
Slide of 13
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
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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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Product of Powers Property
x^2y^3 (xy^2)
CommutativePropMult
Commutative Property of Multiplication
x^2y^3 (y^2x)
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^3y^5
Example
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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 = 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
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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.
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.
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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
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^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^7y^2
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Multiplying Monomials
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Quotient of Powers Property
Example
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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.
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^4/x^2
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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.
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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.
x^4/x^2 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^4/x^2 &= 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^4y^3/x^2y^2 = x^4/x^2 * y^3/y^2 Next, use the Quotient of Powers Property for each division. x^4/x^2 * y^3/y^2 &= 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^2y 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!
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Dividing Monomials
Example
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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.
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.
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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.
b This time the given monomial has two variables.
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Evaluating Monomials in One Variable
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Evaluating Monomials in Two Variables
Closure
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Counting the Heads of Garlic
Ignacio was asked to peel all the heads of garlic in the crate.
x^2y^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^3y^4z
Multiply and Divide Monomials
Exercise 3.1
Find the value of x that satisfies the following equation.
2^x * 2^8 = 2^(14)
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Let's take a look at the given equation. 2^x * 2^8 = 2^(14) Let's start by simplifying the left-hand side of the equation. We have a multiplication of two terms with the base 2. This means that we can use the Product of Powers Property to multiply them by adding the exponents of the terms. 2^x * 2^8 &= 2^(14) 2^(x+ 8) &= 2^(14) We have an equation of two terms that have the same base. In order to satisfy the equation, their exponents must also be equal to each other. This means that the sum of x and 8 must be equal to 14. x+ 8= 14 Finally, let's isolate x by subtracting 8 from both sides of the equation.
This means that the given equation is true when x is 6.
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Find the value of x that satisfies the following equation.
5^x/5^2 = 5^7
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Let's take a look at the given equation. 5^x/5^2 = 5^7 We can begin by simplifying the left-hand side of the equation. We have a division of two terms with the base 5. This means that we can use the Quotient of Powers Property to divide them by subtracting the exponents of the terms. 5^x/5^2 &= 5^7 [1em] 5^(x- 2) &= 5^7 We have an equation of two terms that have the same base. In order to satisfy the equation, their exponents must also be equal to each other. This means we get 7 by subtracting 2 from x. x- 2= 7 Let's isolate x by adding 2 to both sides of the equation.
This means that the given equation is true when x is 9.
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Multiply and Divide Monomials
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