S E C T I O N 2.4

Multiplying Integers

Multiplication of integers is very much like multiplication of whole numbers.The only difference is that we must determine whether the answer is positive or negative.

When we multiply two nonzero integers, they either have different signs or they have the same sign. This means that there are two possibilities to consider.

1 Multiply two integers that have different signs.

To develop a rule for multiplying two integers that have different signs, we will find 4( 3), which is the product of a positive integer and negative integer. We say that the signs of the factors are unlike. By the definition of multiplication, 4( 3) means that we are to add 3 four times.

4( 3) ( 3) ( 3) ( 3) ( 3) Write 3 as an addend four times.

12 Use the rule for adding two integers that have the same sign.

The result is negative.As a check, think in terms of money. If you lose $3 four times, you have lost a total of $12,which is written $12.This example illustrates the following rule.

Multiplying Two Integers That Have Different (Unlike) Signs

To multiply a positive integer and a negative integer, multiply their absolute values. Then make the final answer negative.

Strategy We will use the rule for multiplying two integers that have different (unlike) signs.

WHY In each case,we are asked to multiply a positive integer and a negative integer.

d.Recall from Section 1.4, to find the product of a whole number and 10, 100, 1,000, and so on, attach the number of zeros in that number to the right of the whole number. This rule can be extended to products of integers and 10, 100, 1,000, and so on.

34(1,000) 34,000 Since 1,000 has three zeros, attach three 0’s after 34.

Objectives

1Multiply two integers that have different signs.

2Multiply two integers that have the same sign.

3Perform several multiplications to evaluate expressions.

4Evaluate exponential expressions that have negative bases.

5Solve application problems by multiplying integers.

Self Check 1

Multiply:

a.2( 6)

b.30( 4)

c.75 17

d.98(1,000)

Now Try Problems 21,25,29,and 31

EXAMPLE 2 Multiply:

a. 5( 9) b. 8( 10) c. 23( 42) d. 2,500( 30,000)

Strategy We will use the rule for multiplying two integers that have the same (like) signs.

WHY In each case, we are asked to multiply two negative integers.

d.We can extend the method discussed in Section 1.4 for multiplying wholenumber factors with trailing zeros to products of integers with trailing zeros.

2,500( 30,000) 75,000,000 Attach six 0’s after 75.

Multiply 25 and 3 to get 75.

We now summarize the multiplication rules for two integers.

Multiplying Two Integers

To multiply two nonzero integers, multiply their absolute values.

1.The product of two integers that have the same (like) signs is positive.

2.The product of two integers that have different (unlike) signs is negative.

Using Your CALCULATOR Multiplication with Negative Numbers

At Thanksgiving time, a large supermarket chain offered customers a free turkey with every grocery purchase of $200 or more. Each turkey cost the store $8, and 10,976 people took advantage of the offer. Since each of the 10,976 turkeys given away represented a loss of $8 (which can be expressed as $8), the company lost a total of 10,976( $8). To perform this multiplication using a calculator, we enter the following:

The negative result indicates that with the turkey giveaway promotion, the supermarket chain lost $87,808.

3 Perform several multiplications to evaluate expressions.

To evaluate expressions that contain several multiplications, we make repeated use of the rules for multiplying two integers.

Self Check 2

Multiply:

a.9( 7)

b.12( 2)

c.34( 15)

d.4,100( 20,000)

Now Try Problems 33,37,41,and 43

Self Check 3

Evaluate each expression:

a.3( 12)( 2)

b.1(9)( 6)

c.4( 5)(8)( 3)

Now Try Problems 45, 47, and 49

Strategy Since there are no calculations within parentheses and no exponential expressions, we will perform the multiplications, working from the left to the right.

WHY This is step 3 of the order of operations rule that was introduced in Section 1.9.

Solution

b. 9(8)( 1) 72( 1) Use the rule for multiplying two integers that have different

The properties of multiplication that were introduced in Section 1.3, Multiplying Whole Numbers, are also true for integers.

Self Check 4

Use the commutative and/or associative properties of multiplication to evaluate each expression from Self Check 3 in a different way:

a.3( 12)( 2)

b.1(9)( 6)

c.4( 5)(8)( 3)

Now Try Problems 45, 47, and 49

Properties of Multiplication

Commutative property of multiplication: The order in which integers are multiplied does not change their product.

Associative property of multiplication: The way in which integers are grouped does not change their product.

Multiplication property of 0: The product of any integer and 0 is 0.

Multiplication property of 1: The product of any integer and 1 is that integer.

Another approach to evaluate expressions like those in Example 3 is to use the properties of multiplication to reorder and regroup the factors in a helpful way.

Use the commutative and/or associative properties of multiplication to evaluate each expression from Example 3 in a different way:

a. 6( 2)( 7) b. 9(8)( 1) c. 3( 5)(2)( 4)

Strategy When possible, we will use the commutative and/or associative properties of multiplication to multiply pairs of negative factors.

WHY The product of two negative factors is positive.With this approach, we work with fewer negative numbers, and that lessens the possibility of an error.

Solution

b. 9(8)( 1) 9(8) Multiply the negative factors to produce a positive product: 9( 1) 9.

72

Strategy When possible, we will use the commutative and/or associative properties of multiplication to multiply pairs of negative factors.

WHY The product of two negative factors is positive.With this approach, we work with fewer negative numbers, and that lessens the possibility of an error.

Solution

a.Note that this expression is the product of three (an odd number) negative integers.

2( 4)( 5) 8( 5) Multiply the first two negative factors to produce a positive product.

40 The product is negative.

b.Note that this expression is the product of four (an even number) negative integers.

3( 2)( 6)( 5) 6(30) Multiply the first two negative factors and the last two negative factors to produce positive products.

180 The product is positive.

Example 5, part a, illustrates that a product is negative when there is an odd number of negative factors. Example 5, part b, illustrates that a product is positive when there is an even number of negative factors.

Multiplying an Even and an Odd Number of Negative Integers

The product of an even number of negative integers is positive.

The product of an odd number of negative integers is negative.

4 Evaluate exponential expressions that have negative bases.

Recall that exponential expressions are used to represent repeated multiplication. For example, 2 to the third power, or 23, is a shorthand way of writing 2 2 2. In this expression, the exponent is 3 and the base is positive 2. In the next example, we evaluate exponential expressions with bases that are negative numbers.

EXAMPLE 6 Evaluate each expression: a. ( 2)4 b. ( 5)3 c. ( 1)5

Strategy We will write each exponential expression as a product of repeated factors and then perform the multiplication. This requires that we identify the base and the exponent.

WHY The exponent tells the number of times the base is to be written as a factor.

Self Check 5

Evaluate each expression:

a.1( 2)( 5)

b.2( 7)( 1)( 2)

Now Try Problems 53 and 57

Self Check 6

Evaluate each expression:

a.( 3)4

b.( 4)3

c.( 1)7

EXAMPLE 7 Evaluate: 22

Strategy We will rewrite the expression as a product of repeated factors, and then perform the multiplication. We must be careful when identifying the base. It is 2, not 2.

WHY Since there are no parentheses around 2, the base is 2.

Solution

Self Check 7

Evaluate: 42

Now Try Problem 71

Using Your CALCULATOR Raising a Negative Number to a Power

We can find powers of negative integers, such as ( 5)6, using a calculator. The keystrokes that are used to evaluate such expressions vary from model to model, as shown below. You will need to determine which keystrokes produce the positive result that we would expect when raising a negative number to an even power.

From the calculator display, we see that ( 5)6 15,625.

5 Solve application problems by multiplying integers.

Problems that involve repeated addition are often more easily solved using multiplication.

Form If we use negative numbers to represent the depths at which the measurements were made, then the first was at 75 feet. The depth (in feet) of the submersible when the 25th measurement was made can be found by adding 75 twenty-five times. This repeated addition can be calculated more simply by multiplication.

Self Check 8

GASOLINE LEAKS To determine how badly a gasoline tank was leaking, inspectors used a drilling process to take soil samples nearby. The first sample was taken 6 feet below ground level, and more were taken every 6 feet after that. The 14th sample was the first one that did not show signs of gasoline. How far below ground level was that?

Now Try Problem 97

25( 75) 1,875 Multiply the absolute values, 25 and 75, to get 1,875.

Since the integers have different signs, make the final answer negative.

7525 375 1 500 1,875

S E C T I O N 2.4 STUDY SET

VOCABULARY

Fill in the blanks.

1.In the multiplication problem shown below, label each factor and the product.

CONCEPTS

Fill in the blanks.

7.Multiplication of integers is very much like multiplication of whole numbers. The only difference is that we must determine whether the answer is

or .

8. When we multiply two nonzero integers, they either have signs or sign.

9.To multiply a positive integer and a negative integer, multiply their absolute values. Then make the final

answer .

10.To multiply two integers that have the same sign, multiply their absolute values. The final answer is

.

16.If each of the following expressions were evaluated, what would be the sign of the result?

NOTATION

17.For each expression, identify the base and the exponent.

18.Translate to mathematical symbols.

a.negative three times negative two

b.negative five squared

c.the opposite of the square of five

Complete each solution to evaluate the expression.

19. 3( 2)( 4) ( 4)

20.( 3)4 ( 3)( 3)( 3)

(9)

GUIDED PRACTICE

Multiply. See Example 1.

Evaluate each expression. See Examples 3 and 4.

Evaluate each expression. See Example 7.

69.( 7)2 and 72

70.( 5)2 and 52

71.( 12)2 and 122

72.( 11)2 and 112

TRY IT YOURSELF

Evaluate each expression.

83.Find the product of 6 and the opposite of 10.

84.Find the product of the opposite of 9 and the opposite of 8.

95.( 1)( 2)( 3)( 4)( 5)

96.( 10)( 8)( 6)( 4)( 2)

APPLICATIONS

Use signed numbers to solve each problem.

97.SUBMARINES As part of a training exercise, the captain of a submarine ordered it to descend 250 feet, level off for 5 minutes, and then repeat the process several times. If the sub was on the ocean’s surface at the beginning of the exercise, find its depth after the 8th dive.

98.BUILDING A PIER A pile driver uses a heavy weight to pound tall poles into the ocean floor. If each strike of a pile driver on the top of a pole sends it 6 inches deeper, find the depth of the pole after 20 strikes.

Image Source/Getty Images

99.MAGNIFICATION A mechanic used an electronic testing device to check the smog emissions of a car. The results of the test are displayed on a screen.

a.Find the high and low values for this test as shown on the screen.

b.By switching a setting, the picture on the screen can be magnified. What would be the new high and new low if every value were doubled?

100.LIGHT Sunlight is a mixture of all colors. When sunlight passes through water, the water absorbs different colors at different rates, as shown.

a.Use a signed number to represent the depth to which red light penetrates water.

b.Green light penetrates 4 times deeper than red light. How deep is this?

c.Blue light penetrates 3 times deeper than orange light. How deep is this?

Surface of water

101.JOB LOSSES Refer to the bar graph. Find the number of jobs lost in . . .

a.September 2008 if it was about 6 times the number lost in April.

b.October 2008 if it was about 9 times the number lost in May.

c.November 2008 if it was about 7 times the number lost in February.

d.December if it was about 6 times the number lost in March.

2008 U.S. Monthly Net Job Losses

Jan. Feb. Mar. Apr. May June July Aug.

Source: Bureau of Labor Statistics

102.RUSSIA The U.S. Census Bureau estimates that Russia’s population is decreasing by about 700,000 per year because of high death rates and low birth rates. If this pattern continues, what will be the total decline in Russia’s population over the next 30 years? (Source: About.com)

103.PLANETS The average surface temperature of Mars is 81°F. Find the average surface temperature of Uranus if it is four times colder than Mars. (Source:

The World Almanac and Book of Facts, 2009)

104.CROP LOSS A farmer, worried about his fruit trees suffering frost damage, calls the weather service for temperature information. He is told that temperatures will be decreasing approximately

5 degrees every hour for the next five hours. What signed number represents the total change in temperature expected over the next five hours?

105.TAX WRITE-OFF For each of the last six years, a businesswoman has filed a $200 depreciation allowance on her income tax return for an office computer system. What signed number represents the total amount of depreciation written off over the six-year period?

106.EROSION A levee protects a town in a low-lying area from flooding. According to geologists, the banks of the levee are eroding at a rate of 2 feet per year. If something isn’t done to correct the problem, what signed number indicates how much of the levee will erode during the next decade?