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# 7.5: Simplify Products or Quotients of Single Variable Expressions

Difficulty Level: At Grade Created by: CK-12
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Practice Simplify Products or Quotients of Single Variable Expressions
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Have you ever had a stamp collection?

Marc has twice as many stamps in his collection as his Grandfather has in his. Write an expression to represent m\begin{align*}m\end{align*}, the number of stamps in his Grandfather's collection.

To solve this problem, you will need to know how to write a single variable expression. Pay attention to this Concept, and you will know how to do this by the end of the Concept.

### Guidance

Recall that when you add and subtract terms in an expression, you can only combine like terms.

However, you can multiply or divide terms whether they are like terms or not.

For example, 6a\begin{align*}6a\end{align*} and 3a\begin{align*}3a\end{align*} are like terms because both terms include the variable a\begin{align*}a\end{align*}. We can multiply them to simplify an expression like this.

6a×3a=18×a×a=18a2\begin{align*}6a \times 3a= 18 \times a \times a=18a^2\end{align*}.

However, even though 6a\begin{align*}6a\end{align*} and 3 are not like terms, we can still multiply them, like this.

6a×3=18a\begin{align*}6a \times 3=18a\end{align*}.

The Commutative and Associative Properties of Multiplication may help you understand how to multiply expressions with variables. Remember, the Commutative property states that factors can be multiplied in any order. The Associative property states that the grouping of factors does not matter.

Let's apply this information.

6a(3a)\begin{align*}6a(3a)\end{align*}

We can take these two terms and multiply them together.

First, we multiply the number parts.

6×3=18\begin{align*}6 \times 3 = 18\end{align*}

Next, we multiply the variables.

aa=a2\begin{align*}a \cdot a= a^2\end{align*}

Our answer is 18a2\begin{align*}18a^2\end{align*}.

Here is another one.

5x(8y)\begin{align*}5x(8y)\end{align*}

Even though these two terms are different, we can still multiply them together.

First, we multiply the number parts.

5×8=40\begin{align*}5 \times 8 = 40\end{align*}

Next, we multiply the variables.

xy=xy\begin{align*}x \cdot y=xy\end{align*}

Our answer is 40xy\begin{align*}40xy\end{align*}.

Find the product 4z×12\begin{align*}4z \times \frac{1}{2}\end{align*}.

4z\begin{align*}4z\end{align*} and 12\begin{align*}\frac{1}{2}\end{align*} are not like terms, however, you can multiply terms even if they are not like terms.

Use the commutative and associative properties to rearrange the factors to make it easier to see how they can be multiplied.

According to the commutative property, the order of the factors does not matter.

So, 4z×12=12×4z\begin{align*}4z \times \frac{1}{2}=\frac{1}{2}\times 4z\end{align*}.

According to the associative property, the grouping of the factors does not matter. Group the factors so that the numbers are multiplied first.

So, 12×4z=12×4×z=(12×4)×z\begin{align*}\frac{1}{2} \times 4z=\frac{1}{2} \times 4 \times z=\left(\frac{1}{2} \times 4\right) \times z\end{align*}.

Now, multiply.

(12×4)×z=(12×41)×z=42×z=2×z=2z.\begin{align*}\left(\frac{1}{2} \times 4\right) \times z=\left(\frac{1}{2} \times \frac{4}{1}\right) \times z=\frac{4}{2}\times z=2 \times z=2z.\end{align*}

The product is 2z\begin{align*}2z\end{align*}.

Remember that the word PRODUCT means multiplication and the word QUOTIENT means division.

Here is one that uses division.

Find the quotient 42c÷7\begin{align*}42c \div 7\end{align*}.

It may help you to rewrite the problem like this 42c7\begin{align*}\frac{42c}{7}\end{align*}. Then separate out the numbers and variables like this.

42c7=42c7=427c\begin{align*}\frac{42c}{7}=\frac{42 \cdot c}{7}=\frac{42}{7} \cdot c\end{align*}

Now, divide 42 by 7 to find the quotient.

427c=6c=6c\begin{align*}\frac{42}{7} \cdot c=6 \cdot c=6c\end{align*}

The quotient is 6c\begin{align*}6c\end{align*}.

Now it's your turn. Find each product or quotient.

#### Example A

6a(9a)\begin{align*}6a(9a)\end{align*}

Solution:54a2\begin{align*}54a^2\end{align*}

#### Example B

15b5b\begin{align*}\frac{15b}{5b}\end{align*}

Solution:3\begin{align*}3\end{align*}

#### Example C

20c4\begin{align*}\frac{20c}{4}\end{align*}

Solution:5c\begin{align*}5c\end{align*}

Here is the original problem once again.

Marc has twice as many stamps in his collection as his Grandfather has in his. Write an expression to represent m\begin{align*}m\end{align*}, the number of stamps in his Grandfather's collection.

To write this, we simply use the variable and the fact that Marc has twice as many stamps.

2m\begin{align*}2m\end{align*}

This term represents Marc's stamps.

### Vocabulary

Here are the vocabulary words in this Concept.

Expression
a number sentence without an equal sign that combines numbers, variables and operations.
Simplify
to make smaller by combining like terms
Product
the answer in a multiplication problem.
Quotient
the answer in a division problem.
Commutative Property of Multiplication
states that the product is not affected by the order in which you multiply factors.
Associative Property of Multiplication
states that the product is not affected by the groupings of the numbers when multiplying.

### Guided Practice

Here is one for you to try on your own.

Find the quotient 50g÷10g\begin{align*}50 g \div 10 g\end{align*}.

It may help you to rewrite the problem like this 50g10g\begin{align*}\frac{50g}{10g}\end{align*}. Then separate out the numbers and variables like this.

50g10g=50g10g=5010gg\begin{align*}\frac{50 g}{10 g}=\frac{50 \cdot g}{10 \cdot g}=\frac{50}{10} \cdot \frac{g}{g}\end{align*}

Now, divide 50 by 10 and divide g\begin{align*}g\end{align*} by g\begin{align*}g\end{align*} to find the quotient. Since any number over itself is equal to 1, you know that gg=1\begin{align*}\frac{g}{g}=1\end{align*}.

5010gg=5.1=5\begin{align*}\frac{50}{10} \cdot \frac{g}{g}=5.1=5\end{align*}

The quotient is 5.

### Video Review

Here is a video for review.

### Practice

Directions: Simplify each product or quotient.

1. 6a(4a)\begin{align*}6a(4a)\end{align*}

2. 9x(2)\begin{align*}9x(2)\end{align*}

3. 14y(2y)\begin{align*}14y(2y)\end{align*}

4. 16a(a)\begin{align*}16a(a)\end{align*}

5. 22x(2x)\begin{align*}22x(2x)\end{align*}

6. 18b(2)\begin{align*}18b(2)\end{align*}

7. 21a7\begin{align*}\frac{21a}{7}\end{align*}

8. 22b2b\begin{align*}\frac{22b}{2b}\end{align*}

9. \begin{align*}\frac{25x}{x}\end{align*}

10. \begin{align*}\frac{45a}{5a}\end{align*}

11. \begin{align*}\frac{15x}{3x}\end{align*}

12. \begin{align*}\frac{18y}{9}\end{align*}

13. \begin{align*}\frac{22y}{11y}\end{align*}

14. \begin{align*}\frac{15x}{3y}\end{align*}

15. \begin{align*}\frac{82x}{2x}\end{align*}

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