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Pre-Algebra View details

8. Multiply and Divide Monomials

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eCourses /

Pre-Algebra /

Chapter 4

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.

Lesson Settings & Tools

	13 Theory slides
	11 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

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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.

Powers
Algebraic Expressions
Monomials
Degree of a Monomial
Associative Property of Multiplication
Commutative Property of Multiplication

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.

The crate contains eight bags of garlic. Each bag has nine heads of garlic. How many heads of garlic does Ignacio have to peel?

Now suppose each crate has

x^{2} y

bags of garlic, and each bag has

y^{3}

bulbs. Write an algebraic expression for the total number of heads of garlic in

x z

many crates.

Discussion

Product of Powers Property

The product of two powers with the same non-zero base $a$ and integer exponents $m$ and $n$ can be written as a single power with base $a$ and exponent $m + n .$

$a^{m} \cdot a^{n} = a^{m + n}$

The Product of Powers Property is useful for multiplying monomials.

x^{2} \cdot x^{3} ⇓ x^{2} \cdot 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} (x y^{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

$a^{m} \cdot a^{n} = a^{m + n}$

x^{2} (y^{5}) (x)

CommutativePropMult

Commutative Property of Multiplication

x^{2} (x) (y^{5})

MultPow

$a^{m} \cdot a^{n} = a^{m + n}$

x^{3} (y^{5})

Multiply

x^{3} y^{5}

Please note that the Product of Powers Property only applies to factors with the same base.

Example

Making Hand-Pulled Noodles

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

A bowl of ramen that looks tasty, with chopsticks holding some 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 .

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

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 \times 2 = 2^{2}

Ms. Ley told Ignacio to fold and stretch the noodles

four

more times.

2 \times 2 \times 2 \times 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} \times 2^{4}

The Product of Powers Property can be used to add the powers.

2^{2} \times 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 \times 2 \times 2 \times 2 \times 2 \times 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.

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 \cdot x \cdot 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} \cdot x^{4} \cdot x

a What is the degree of the monomial?

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) \cdot x^{4} \cdot (x y)

What is the degree of the monomial for his total bonus for the day?

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

a Ignacio got

two

compliments from Table

1,

four

compliments from Table

2,

and

one

compliment from Table

3 .

Rewrite the product. Remember that when a variable has no visible exponent, the exponent is assumed to be

1 .

x^{2} \cdot x^{4} \cdot x ⇓ x^{2} \cdot x^{4} \cdot x^{1}

Use the Product of Powers Property to multiply the monomials by adding the exponents.

x^{2} \cdot x^{4} \cdot 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) \cdot x^{4} \cdot (x y)

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) \cdot x^{4} \cdot (x y) = (x^{2} \cdot y \cdot x^{4}) \cdot x y

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} \cdot y \cdot x^{4}) \cdot (x y) = (x^{2} \cdot x^{4} \cdot y) \cdot x y = (x^{6} \cdot y) \cdot x y

Finish the simplification by combining the rest of the variables in a similar fashion.

(x^{6} \cdot y) \cdot x y

AssociativePropMult

Associative Property of Multiplication

x^{6} \cdot (y \cdot x \cdot y)

CommutativePropMult

Commutative Property of Multiplication

x^{6} \cdot (x \cdot y \cdot y)

ProdToPowTwoFac

$a \cdot a = a^{2}$

x^{6} \cdot (x \cdot y^{2})

AssociativePropMult

Associative Property of Multiplication

(x^{6} \cdot x) \cdot y^{2}

MultPow

$a^{m} \cdot a^{n} = a^{m + n}$

x^{7} \cdot y^{2}

Multiply

x^{7} y^{2}

Typically, monomials with multiple variables are not written with multiplication symbols. The degree of the monomial with several variables is the sum of the degrees of each variable.

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

Multiply the given monomials. Write the degree of the resulting monomial.

Discussion

Quotient of Powers Property

The ratio of two powers with the same non-zero base $a$ and integer exponents $m$ and $n$ can be written as a single power with base $a$ and exponent $m - n .$

$\frac{a ^{m}}{a ^{n}} = a^{m - n}$

This property is particularly useful for the division of monomials.

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.

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.

\frac{x ^{4}}{x ^{2}}

a What is the degree of the monomial resulting from the above quotient?

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.

\frac{x ^{4} y ^{3}}{x ^{2} y ^{2}}

What is the degree of the monomial resulting from the above quotient?

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.

\frac{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.

\frac{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!

b To find the degree of the quotient, begin by splitting the quotient into a product of two fractions. One factor will have the variable

x

and the other factor the variable

y .

\frac{x ^{4} y ^{3}}{x ^{2} y ^{2}} = \frac{x ^{4}}{x ^{2}} \cdot \frac{y ^{3}}{y ^{2}}

Next, use the Quotient of Powers Property for each division.

\frac{x ^{4}}{x ^{2}} \cdot \frac{y ^{3}}{y ^{2}} = x^{4 - 2} \cdot y^{3 - 2} = x^{2} \cdot y^{1}

Any number raised to the power of

1

is equal to itself.

x^{2} \cdot y^{1} = x^{2} \cdot y

Finally, multiply the monomials.

x^{2} \cdot 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

Dividing Monomials

Find the quotient of the given monomials. Write the degree of the resulting monomial.

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.

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.

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.

The following monomial represents the combination to the safe.

m^{3} d^{2}

Today is March

1 0^{th} .

This corresponds to

m = 3

and

d = 10 .

Evaluate the monomial for the given values to find the combination to the safe.

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.

x^{2}

Substitute

4

for

x

to evaluate the monomial.

x^{2}

Substitute

$x = 4$

4^{2}

CalcPow

Calculate power

16

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} \cdot 10^{2}

CalcPow

Calculate power

27 \cdot 100

Multiply

2700

This means that today's combination is

2700 .

Now Ignacio can start working on the secret sauce!

Pop Quiz

Evaluating Monomials in One Variable

Evaluate the monomial for the given value of the variable.

Pop Quiz

Evaluating Monomials in Two Variables

Evaluate the monomial for the given values of the variables.

Closure

Counting the Heads of Garlic

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 \times 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 \cdot y^{3} = x^{2} y^{4}

There are

x z

many crates, so this monomial needs to be multiplied by

x z .

x^{2} y^{4} \cdot x z

AssociativePropMult

Associative Property of Multiplication

x^{2} \cdot (y^{4} \cdot x) \cdot z

CommutativePropMult

Commutative Property of Multiplication

x^{2} \cdot (x \cdot y^{4}) \cdot z

AssociativePropMult

Associative Property of Multiplication

(x^{2} \cdot x) \cdot (y^{4} \cdot z)

MultPow

$a^{m} \cdot a^{n} = a^{m + n}$

x^{3} \cdot y^{4} \cdot z

Multiply

x^{3} y^{4} z

The total number of heads of garlic in all the crates is expressed by the monomial

x^{3} y^{4} z .

Level 1

Level 2

Level 3

Multiply and Divide Monomials

Exercises

Write each expression as a monomial using exponents.

a \cdot a \cdot a \cdot a \cdot a \cdot a

b \cdot b \cdot b

c \cdot c \cdot c \cdot c

Let's take a look at the given expression. a * a * a * a * a * a This expression has a repeated factor a. It is multiplied by itself six times. Recall that a monomial can be written by taking the repeated factor as a base and the number of times that the factor is multiplied by itself as an exponent. This means that we can write the given expression as a monomial as a to the power of 6. a * a * a * a * a * a = a^6

This time the given expression has a repeated factor b. It is multiplied by itself three times. b * b * b This corresponds to b raised to the power of 3. b * b * b = b^3

The last expression is a value c multiplied by itself four times. c * c * c * c This corresponds to c raised to the power of 4. c * c * c * c = c^4

Write each expression as a monomial using exponents.

a \cdot a \cdot b \cdot b \cdot a \cdot a

x \cdot y \cdot x \cdot y

Let's take a look at the given expression. a * a * b * b * a * a Notice that this expression has two repeated factors, a and b. The factor a is repeated four times and the factor b is repeated two times. This means that we can write the given expression as a monomial as the product of the variable a raised to the power of 4 and the variable b raised to the power of 2. a * a * b * b * a * a = a^4b^2 Because of the Commutative Property of Multiplication, b^2a^4 is also a possible answer.

Let's take a look at the next expression. x * y * x * y This expression has x two times as a factor and y two times as a factor. This means that we can write it as a monomial just as we did on Part A. x * y * x * y = x^2y^2 Please note that y^2x^2 is a possible answer as well.

Multiply the monomials.

p^{4} \cdot p^{7}

x^{8} \cdot x

z^{2} \cdot z^{2} \cdot z^{4}

Let's take a look at the given expression. p^4 * p^7 The Product of Powers Property tells us that when we multiply monomials with the same non-zero base, we can add their powers. In our expression, we have two monomials with the base p. Let's use the Product of Powers Property to multiply them by adding the powers of our monomials. p^4 * p^7 &= p^(4+ 7) &= p^(11)

Let's take a look at the next expression. x^8 * x When a variable has no power written over it, the power is assumed to be 1. x = x^1 We can rewrite the given expression by using this information. x^8 * x^1 Let's now add the powers of the monomials x^8 and x^1 by the Product of Powers Property. x^8 * x^1 &= x^(8+ 1) &= x^9

Let's take a look at the last expression. z^2 * z^2 * z^4 This time we have three monomials. The base of all three monomials is z, so we can use the Products of Powers Property to multiply them. We just need to add the three powers! z^2 * z^2 * z^4 &= z^(2+2+4) &= z^8

Divide the monomials.

\frac{p ^{8}}{p ^{4}}

\frac{q ^{5}}{q ^{4}}

Let's take a look at the given expression. p^8/p^4 The Quotient of Powers Property tells us that when we divide monomials with the same non-zero base, we subtract their powers. In our expression, we have two monomials each with the base p. Let's use the Quotient of Powers Property to divide them. We do this by subtracting the power of the denominator from the power of the numerator. p^8/p^4 &= p^(8- 4) &= p^4

Let's take a look at the next expression. q^5/q^4 In this expression, we have the division of two monomials with the base q. We will once again subtract the power of the denominator from the power of the numerator by using the Quotient of Powers Property. q^5/q^4 &= q^(5- 4) &= q^1 Any number raised to the power of 1 is equal to itself. q^1 = q Therefore, we can also express our result as q. q^5/q^4 = q

Multiply and divide the monomials as required. Write the resulting expression in its simplest form.

\frac{x ^{6} \cdot x ^{3}}{x ^{5}}

\frac{z ^{9}}{z ^{2} \cdot z}

Let's take a look at the given expression. x^6 * x^3/x^5 Let's begin by multiplying the monomials in the numerator. Notice that they have the same base x. This means that we can use the Product of Powers Property to multiply them by adding the powers of the monomials. x^6 * x^3/x^5 &= x^(6+ 3)/x^5 [0.75em] &= x^9/x^5 The resulting monomials have the same base x. Now we can use the Quotient of Powers Property to divide them by subtracting the power of the denominator from the power of the numerator. x^9/x^5 &= x^(9- 5) [0.5em] &= x^4 The result is x^4.

Let's take a look at the next expression. z^9/z^2* z We can start by multiplying the monomials in the denominator. Notice that they have the same base z. Let's use the Product of Powers Property to multiply them by adding the powers of the monomials. Do not forget that if no power is shown, it is assumed to be 1. z^9/z^2* z &= z^9/z^(2+ 1) [o.75em] &= z^9/z^3 The resulting monomials have the same base z. Let's use the Quotient of Powers Property again to subtract the power of the denominator from the power of the numerator. z^9/z^3 &= z^(9- 3) [0.5em] &= z^6 The result is z^6.

Write the degree of the given monomial.

x^{3}

x^{2} y z^{4}

Let's take a look at the given monomial. x^3 The power of this monomial is 3. This means that x is being multiplied by itself 3 times. x^3 = x * x * x Since there are 3 factors present in the above monomial, its degree is 3.

Let's take a look at the next monomial. x^2yz^4 This monomial has three variables. The x-variable is present as a factor 2 times, the y-variable just 1 time, and the z-variable 4 times. x^2y^1z^4 = x * x * y * z * z * z * z Add the number of times each variable is being used as a factor. The result will give us the degree of the monomial. 2+ 1+ 4 = 7 Another way of finding the degree of the monomial is calculating the sum of the powers of the variables. Keep in mind that when a variable has no power written over it, it is assumed that its power is 1. x^2yz^4 = x^2 y^1 z^4 ⇓ 2+ 1+ 4 = 7 This means that the degree of the given monomial is 7.

Multiply and Divide Monomials

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