Learning Objectives
By the end of this section, you will be able to:
- Multiply integers
- Divide integers
- Simplify expressions with integers
- Evaluate variable expressions with integers
- Translate word phrases to algebraic expressions
Be Prepared 3.7
Before you get started, take this readiness quiz.
Translate the quotient of and into an algebraic expression.
If you missed this problem, review Example 1.67.
Be Prepared 3.8
Add:
If you missed this problem, review Example 3.21.
Be Prepared 3.9
If you missed this problem, review Example 3.23.
Multiply Integers
Since multiplication is mathematical shorthand for repeated addition, our counter model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction.
We remember that means add times. Here, we are using the model shown in Figure 3.19 just to help us discover the pattern.
Now consider what it means to multiply by It means subtract times. Looking at subtraction as taking away, it means to take away times. But there is nothing to take away, so we start by adding neutral pairs as shown in Figure 3.20.
In both cases, we started with neutral pairs. In the case on the left, we took away times and the result was To multiply we took away times and the result was So we found that
Notice that for multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative.
Multiplication of Signed Numbers
The sign of the product of two numbers depends on their signs.
Same signs | Product |
---|---|
•Two positives •Two negatives |
Positive Positive |
Different signs | Product |
---|---|
•Positive • negative •Negative • positive |
Negative Negative |
Example 3.47
Multiply each of the following:
- ⓐ
- ⓑ
- ⓒ
- ⓓ
Solution
ⓐ | |
Multiply, noting that the signs are different and so the product is negative. |
ⓑ | |
Multiply, noting that the signs are the same and so the product is positive. |
ⓒ | |
Multiply, noting that the signs are different and so the product is negative. |
ⓓ | |
The signs are the same, so the product is positive. |
Try It 3.93
Multiply:
- ⓐ
- ⓑ
- ⓒ
- ⓓ
Try It 3.94
Multiply:
- ⓐ
- ⓑ
- ⓒ
- ⓓ
When we multiply a number by the result is the same number. What happens when we multiply a number by Let’s multiply a positive number and then a negative number by to see what we get.
Each time we multiply a number by we get its opposite.
Multiplication by
Multiplying a number by gives its opposite.
Example 3.48
Multiply each of the following:
- ⓐ
- ⓑ
Solution
ⓐ | |
The signs are different, so the product will be negative. | |
Notice that −7 is the opposite of 7. |
ⓑ | |
The signs are the same, so the product will be positive. | |
Notice that 11 is the opposite of −11. |
Try It 3.95
Multiply.
- ⓐ
- ⓑ
Try It 3.96
Multiply.
- ⓐ
- ⓑ
Divide Integers
Division is the inverse operation of multiplication. So, because In words, this expression says that can be divided into groups of each because adding five three times gives If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.
Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.
Division of Signed Numbers
The sign of the quotient of two numbers depends on their signs.
Same signs | Quotient |
---|---|
•Two positives •Two negatives |
Positive Positive |
Different signs | Quotient |
---|---|
•Positive & negative •Negative & positive |
Negative Negative |
Remember, you can always check the answer to a division problem by multiplying.
Example 3.49
Divide each of the following:
- ⓐ
- ⓑ
Solution
ⓐ | |
Divide, noting that the signs are different and so the quotient is negative. |
ⓑ | |
Divide, noting that the signs are the same and so the quotient is positive. |
Try It 3.97
Divide:
- ⓐ
- ⓑ
Try It 3.98
Divide:
- ⓐ
- ⓑ
Just as we saw with multiplication, when we divide a number by the result is the same number. What happens when we divide a number by Let’s divide a positive number and then a negative number by to see what we get.
When we divide a number by, we get its opposite.
Division by
Dividing a number by gives its opposite.
Example 3.50
Divide each of the following:
- ⓐ
- ⓑ
Solution
ⓐ | |
The dividend, 16, is being divided by –1. | |
Dividing a number by –1 gives its opposite. | |
Notice that the signs were different, so the result was negative. |
ⓑ | |
The dividend, –20, is being divided by –1. | |
Dividing a number by –1 gives its opposite. |
Notice that the signs were the same, so the quotient was positive.
Try It 3.99
Divide:
- ⓐ
- ⓑ
Try It 3.100
Divide:
- ⓐ
- ⓑ
Simplify Expressions with Integers
Now we’ll simplify expressions that use all four operations–addition, subtraction, multiplication, and division–with integers. Remember to follow the order of operations.
Example 3.51
Solution
We use the order of operations. Multiply first and then add and subtract from left to right.
Multiply first. | |
Add. | |
Subtract. |
Try It 3.101
Simplify:
Try It 3.102
Simplify:
Example 3.52
Simplify:
- ⓐ
- ⓑ
Solution
The exponent tells how many times to multiply the base.
ⓐ The exponent is and the base is We raise to the fourth power.
Write in expanded form. | |
Multiply. | |
Multiply. | |
Multiply. |
ⓑ The exponent is and the base is We raise to the fourth power and then take the opposite.
Write in expanded form. | |
Multiply. | |
Multiply. | |
Multiply. |
Try It 3.103
Simplify:
- ⓐ
- ⓑ
Try It 3.104
Simplify:
- ⓐ
- ⓑ
Example 3.53
Solution
According to the order of operations, we simplify inside parentheses first. Then we will multiply and finally we will subtract.
Subtract the parentheses first. | |
Multiply. | |
Subtract. |
Try It 3.105
Simplify:
Try It 3.106
Simplify:
Example 3.54
Simplify:
Solution
We simplify the exponent first, then multiply and divide.
Simplify the exponent. | |
Multiply. | |
Divide. |
Try It 3.107
Simplify:
Try It 3.108
Simplify:
Example 3.55
Solution
First we will multiply and divide from left to right. Then we will add.
Divide. | |
Multiply. | |
Add. |
Try It 3.109
Simplify:
Try It 3.110
Simplify:
Evaluate Variable Expressions with Integers
Now we can evaluate expressions that include multiplication and division with integers. Remember that to evaluate an expression, substitute the numbers in place of the variables, and then simplify.
Example 3.56
Solution
Simplify exponents. | |
Multiply. | |
Subtract. | |
Add. |
Keep in mind that when we substitute for we use parentheses to show the multiplication. Without parentheses, it would look like
Try It 3.111
Evaluate:
Try It 3.112
Evaluate:
Example 3.57
Solution
Substitute and . | |
Multiply. | |
Simplify. |
Try It 3.113
Evaluate:
Try It 3.114
Evaluate:
Translate Word Phrases to Algebraic Expressions
Once again, all our prior work translating words to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is product and for division is quotient.
Example 3.58
Translate to an algebraic expression and simplify if possible: the product of and
Solution
The word product tells us to multiply.
the product of and | |
Translate. | |
Simplify. |
Try It 3.115
Translate to an algebraic expression and simplify if possible:
Try It 3.116
Translate to an algebraic expression and simplify if possible:
Example 3.59
Translate to an algebraic expression and simplify if possible: the quotient of and
Solution
The word quotient tells us to divide.
the quotient of −56 and −7 | |
Translate. | |
Simplify. |
Try It 3.117
Translate to an algebraic expression and simplify if possible:
Try It 3.118
Translate to an algebraic expression and simplify if possible:
Section 3.4 Exercises
Practice Makes Perfect
Multiply Integers
In the following exercises, multiply each pair of integers.
Divide Integers
In the following exercises, divide.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
- ⓐ
- ⓑ
- ⓐ
- ⓑ
when
when
when and
when and
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate to an algebraic expression and simplify if possible.
The product of and
The quotient of and
The quotient of and the sum of and
The product of and the difference of
Everyday Math
Stock market Javier owns shares of stock in one company. On Tuesday, the stock price dropped per share. What was the total effect on Javier’s portfolio?
Weight loss In the first week of a diet program, eight women lost an average of each. What was the total weight change for the eight women?
Writing Exercises
In your own words, state the rules for dividing two integers.
Why is
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?