Mastering Monomials: Multiplication, Division, and Applying the Product of Powers Property

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!

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Catch-Up and Review

How Many Heads of Garlic?

Product of Powers Property

Making Hand-Pulled Noodles

Hint

Solution

Great Job, Ignacio!

Hint

Solution

Multiplying Monomials

Quotient of Powers Property

Be Careful with Customers

Hint

Solution

Dividing Monomials

Making Dumplings

Hint

Solution

Evaluating Monomials in One Variable

Evaluating Monomials in Two Variables

Counting the Heads of Garlic

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