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3.5: Multiply and Divide Integers

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    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 2020 and 1313 into an algebraic expression.
    If you missed this problem, review Example 1.67.

    Be Prepared 3.8

    Add: −5+(−5)+(−5).−5+(−5)+(−5).
    If you missed this problem, review Example 3.21.

    Be Prepared 3.9

    Evaluaten+4whenn=−7.Evaluaten+4whenn=−7.
    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 a·bFigure 3.19 just to help us discover the pattern.

    This image has two columns. The first column has 5 times 3. Underneath, it states add 5, 3 times. Under this there are 3 rows of 5 blue circles labeled 15 positives and 5 times 3 equals 15. The second column has negative 5 times 3. Underneath it states add negative 5, 3 times. Under this there are 3 rows of 5 red circles labeled 15 negatives and negative 5 times 3 equals 15.
    Figure 3.19

    Now consider what it means to multiply 5Figure 3.20.

    This figure has 2 columns. The first column has 5 times negative 3. Underneath it states take away 5, 3 times. Under this there are 3 rows of 5 red circles. A downward arrow points to six rows of alternating colored circles in rows of fives. The first row includes 5 red circles, followed by five blue circles, then 5 red, five blue, five red, and five blue. All of the rows of blue circles are circled. The non-circled rows are labeled 15 negatives. Under the label is 5 times negative 3 equals negative 15. The second column has negative 5 times negative 3. Underneath it states take away negative 5, 3 times. Then there are 6 rows of 5 circles alternating in color. The first row is 5 blue circles followed by 5 red circles. All of the red rows are circled. The non-circles rows are labeled 15 positives. Under the label is negative 5 times negative 3 equals 15.
    Figure 3.20

    In both cases, we started with 1515 neutral pairs. In the case on the left, we took away 5,35,3 times and the result was −15.−15. To multiply (−5)(−3),(−5)(−3), we took away −5,3−5,3 times and the result was 15.15. So we found that

    5·3=15−5(3)=−155(−3)=−15(−5)(−3)=155·3=15−5(3)=−155(−3)=−15(−5)(−3)=15

    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:

    1. ⓐ−9·3 −9·3
    2. ⓑ−2(−5)−2(−5)
    3. ⓒ4(−8)4(−8)
    4. ⓓ7·67·6
    Answer

    ⓐ
      –9⋅3–9⋅3
    Multiply, noting that the signs are different and so the product is negative. –27–27
    Table 3.5
    ⓑ
      –2(–5)–2(–5)
    Multiply, noting that the signs are the same and so the product is positive. 1010
    Table 3.6
    ⓒ
      4(–8)4(–8)
    Multiply, noting that the signs are different and so the product is negative. –32–32
    Table 3.7
    ⓓ
      7⋅67⋅6
    The signs are the same, so the product is positive. 4242
    Table 3.8

    Try It 3.93

    Multiply:

    1. ⓐ−6·8−6·8
    2. ⓑ−4(−7)−4(−7)
    3. ⓒ9(−7)9(−7)
    4. ⓓ5·125·12

    Try It 3.94

    Multiply:

    1. ⓐ−8·7−8·7
    2. ⓑ−6(−9)−6(−9)
    3. ⓒ7(−4)7(−4)
    4. ⓓ3·133·13

    When we multiply a number by 1,1, the result is the same number. What happens when we multiply a number by−1?−1? Let’s multiply a positive number and then a negative number by −1−1 to see what we get.

    −1·4−1(−3)−43−4is the opposite of43is the opposite of−3−1·4−1(−3)−43−4is the opposite of43is the opposite of−3

    Each time we multiply a number by −1,−1, we get its opposite.

    Multiplication by −1 −1

    Multiplying a number by −1−1 gives its opposite.

    −1a=−a−1a=−a

    Example 3.48

    Multiply each of the following:

    1. ⓐ−1·7−1·7
    2. ⓑ−1(−11)−1(−11)
    Answer

    ⓐ
    The signs are different, so the product will be negative. −1⋅7−1⋅7
    Notice that −7 is the opposite of 7. −7−7
    ⓑ
    The signs are the same, so the product will be positive. −1(−11)−1(−11)
    Notice that 11 is the opposite of −11. 1111

    Try It 3.95

    Multiply.

    1. ⓐ−1·9−1·9
    2. ⓑ−1·(−17)−1·(−17)

    Try It 3.96

    Multiply.

    1. ⓐ−1·8−1·8
    2. ⓑ−1·(−16)−1·(−16)

    Divide Integers

    Division is the inverse operation of multiplication. So, 15÷3=515÷3=5 because 5·3=155·3=15 In words, this expression says that 1515 can be divided into 33 groups of 55 each because adding five three times gives 15.15. If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.

    5·3=15so15÷3=5−5(3)=−15so−15÷3=−5(−5)(−3)=15so15÷(−3)=−55(−3)=−15so−15÷−3=55·3=15so15÷3=5−5(3)=−15so−15÷3=−5(−5)(−3)=15so15÷(−3)=−55(−3)=−15so−15÷−3=5

    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:

    1. ⓐ−27÷3 −27÷3
    2. ⓑ−100÷(−4)−100÷(−4)
    Answer

    ⓐ
      –27÷3–27÷3
    Divide, noting that the signs are different and so the quotient is negative. –9–9
    Table 3.9
    ⓑ
      –100÷(–4)–100÷(–4)
    Divide, noting that the signs are the same and so the quotient is positive. 2525
    Table 3.10

    Try It 3.97

    Divide:

    1. ⓐ−42÷6−42÷6
    2. ⓑ−117÷(−3)−117÷(−3)

    Try It 3.98

    Divide:

    1. ⓐ−63÷7−63÷7
    2. ⓑ−115÷(−5)−115÷(−5)

    Just as we saw with multiplication, when we divide a number by 1,1, the result is the same number. What happens when we divide a number by −1?−1? Let’s divide a positive number and then a negative number by −1−1 to see what we get.

    8÷(−1)−9÷(−1)−89−8 is the opposite of 89 is the opposite of −98÷(−1)−9÷(−1)−89−8 is the opposite of 89 is the opposite of −9

    When we divide a number by, −1−1 we get its opposite.

    Division by −1 −1

    Dividing a number by −1−1 gives its opposite.

    a÷(−1)=−aa÷(−1)=−a

    Example 3.50

    Divide each of the following:

    1. ⓐ16÷(−1)16÷(−1)
    2. ⓑ−20÷(−1)−20÷(−1)
    Answer

    ⓐ
      16÷(–1)16÷(–1)
    The dividend, 16, is being divided by –1. –16–16
    Dividing a number by –1 gives its opposite.  
    Notice that the signs were different, so the result was negative.  
    Table 3.11
    ⓑ
      –20÷(–1)–20÷(–1)
    The dividend, –20, is being divided by –1. 2020
    Dividing a number by –1 gives its opposite.  
    Table 3.12

    Notice that the signs were the same, so the quotient was positive.

    Try It 3.99

    Divide:

    1. ⓐ 6÷(−1) 6÷(−1)
    2. ⓑ −36÷(−1)−36÷(−1)

    Try It 3.100

    Divide:

    1. ⓐ 28÷(−1) 28÷(−1)
    2. ⓑ −52÷(−1)−52÷(−1)

    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

    Simplify:7(−2)+4(−7)−6.Simplify:7(−2)+4(−7)−6.

    Answer

    We use the order of operations. Multiply first and then add and subtract from left to right.

    7(−2)+4(−7)−67(−2)+4(−7)−6
    Multiply first. −14+(−28)−6−14+(−28)−6
    Add. −42−6−42−6
    Subtract. −48−48

    Try It 3.101

    Simplify:

    8(−3)+5(−7)−48(−3)+5(−7)−4

    Try It 3.102

    Simplify:

    9(−3)+7(−8)−19(−3)+7(−8)−1

    Example 3.52

    Simplify:

    1. ⓐ (−2)4 (−2)4
    2. ⓑ −24−24
    Answer

    The exponent tells how many times to multiply the base.

    ⓐ The exponent is 44 and the base is −2.−2. We raise −2−2 to the fourth power.

    (−2)4(−2)4
    Write in expanded form. (−2)(−2)(−2)(−2)(−2)(−2)(−2)(−2)
    Multiply. 4(−2)(−2)4(−2)(−2)
    Multiply. −8(−2)−8(−2)
    Multiply. 1616

    ⓑ The exponent is 44 and the base is 2.2. We raise 22 to the fourth power and then take the opposite.

    −24−24
    Write in expanded form. −(2⋅2⋅2⋅2)−(2⋅2⋅2⋅2)
    Multiply. −(4⋅2⋅2)−(4⋅2⋅2)
    Multiply. −(8⋅2)−(8⋅2)
    Multiply. −16−16

    Try It 3.103

    Simplify:

    1. ⓐ (−3)4 (−3)4
    2. ⓑ −34−34

    Try It 3.104

    Simplify:

    1. ⓐ (−7)2 (−7)2
    2. ⓑ −72−72

    Example 3.53

    Simplify:12−3(9−12).Simplify:12−3(9−12).

    Answer

    According to the order of operations, we simplify inside parentheses first. Then we will multiply and finally we will subtract.

    12−3(9−12)12−3(9−12)
    Subtract the parentheses first. 12−3(−3)12−3(−3)
    Multiply. 12−(−9)12−(−9)
    Subtract. 2121

    Try It 3.105

    Simplify:

    17−4(8−11)17−4(8−11)

    Try It 3.106

    Simplify:

    16−6(7−13)16−6(7−13)

    Example 3.54

    Simplify: 8(−9)÷(−2)3.8(−9)÷(−2)3.

    Answer

    We simplify the exponent first, then multiply and divide.

    8(−9)÷(−2)38(−9)÷(−2)3
    Simplify the exponent. 8(−9)÷(−8)8(−9)÷(−8)
    Multiply. −72÷(−8)−72÷(−8)
    Divide. 99

    Try It 3.107

    Simplify:

    12(−9)÷(−3)312(−9)÷(−3)3

    Try It 3.108

    Simplify:

    18(−4)÷(−2)318(−4)÷(−2)3

    Example 3.55

    Simplify:−30÷2+(−3)(−7).Simplify:−30÷2+(−3)(−7).

    Answer

    First we will multiply and divide from left to right. Then we will add.

    −30÷2+(−3)(−7)−30÷2+(−3)(−7)
    Divide. −15+(−3)(−7)−15+(−3)(−7)
    Multiply. −15+21−15+21
    Add. 66

    Try It 3.109

    Simplify:

    −27÷3+(−5)(−6)−27÷3+(−5)(−6)

    Try It 3.110

    Simplify:

    −32÷4+(−2)(−7)−32÷4+(−2)(−7)

    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

    Evaluate2x2−3x+8whenx=−4.Evaluate2x2−3x+8whenx=−4.

    Answer

    .
    . .
    Simplify exponents. .
    Multiply. .
    Subtract. .
    Add. .

    Keep in mind that when we substitute −4−4 for x,x, we use parentheses to show the multiplication. Without parentheses, it would look like 2·−42−3·−4+8.2·−42−3·−4+8.

    Try It 3.111

    Evaluate:

    3x2−2x+6whenx=−33x2−2x+6whenx=−3

    Try It 3.112

    Evaluate:

    4x2−x−5whenx=−24x2−x−5whenx=−2

    Example 3.57

    Evaluate3x+4y−6whenx=−1andy=2.Evaluate3x+4y−6whenx=−1andy=2.

    Answer

    .
    Substitute x=−1x=−1 and y=2y=2. .
    Multiply. .
    Simplify. .

    Try It 3.113

    Evaluate:

    7x+6y−12whenx=−2andy=37x+6y−12whenx=−2andy=3

    Try It 3.114

    Evaluate:

    8x−6y+13whenx=−3andy=−58x−6y+13whenx=−3andy=−5

    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 −2−2 and 14.14.

    Answer

    The word product tells us to multiply.

    the product of −2−2 and 1414
    Translate. (−2)(14)(−2)(14)
    Simplify. −28−28

    Try It 3.115

    Translate to an algebraic expression and simplify if possible:

    the product of −5 and 12the product of −5 and 12

    Try It 3.116

    Translate to an algebraic expression and simplify if possible:

    the product of 8 and −13the product of 8 and −13

    Example 3.59

    Translate to an algebraic expression and simplify if possible: the quotient of −56−56 and −7.−7.

    Answer

    The word quotient tells us to divide.

    the quotient of −56 and −7
    Translate. −56÷(−7)−56÷(−7)
    Simplify. 88

    Try It 3.117

    Translate to an algebraic expression and simplify if possible:

    the quotient of −63 and −9the quotient of −63 and −9

    Try It 3.118

    Translate to an algebraic expression and simplify if possible:

    the quotient of −72 and −9the quotient of −72 and −9

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    Section 3.4 Exercises

    Practice Makes Perfect

    Multiply Integers

    In the following exercises, multiply each pair of integers.

    211.

    −4 · 8 −4 · 8

    212.

    −3·9−3·9

    213.

    −5 ( 7 ) −5 ( 7 )

    214.

    −8 ( 6 ) −8 ( 6 )

    215.

    −18 ( −2 ) −18 ( −2 )

    216.

    −10 ( −6 ) −10 ( −6 )

    217.

    9 ( −7 ) 9 ( −7 )

    218.

    13 ( −5 ) 13 ( −5 )

    219.

    −1 · 6 −1 · 6

    220.

    −1 · 3 −1 · 3

    221.

    −1 ( −14 ) −1 ( −14 )

    222.

    −1 ( −19 ) −1 ( −19 )

    Divide Integers

    In the following exercises, divide.

    223.

    −24 ÷ 6 −24 ÷ 6

    224.

    −28 ÷ 7 −28 ÷ 7

    225.

    56 ÷ ( −7 ) 56 ÷ ( −7 )

    226.

    35 ÷ ( −7 ) 35 ÷ ( −7 )

    227.

    −52 ÷ ( −4 ) −52 ÷ ( −4 )

    228.

    −84 ÷ ( −6 ) −84 ÷ ( −6 )

    229.

    −180 ÷ 15 −180 ÷ 15

    230.

    −192 ÷ 12 −192 ÷ 12

    231.

    49 ÷ ( −1 ) 49 ÷ ( −1 )

    232.

    62 ÷ ( −1 ) 62 ÷ ( −1 )

    Simplify Expressions with Integers

    In the following exercises, simplify each expression.

    233.

    5 ( −6 ) + 7 ( −2 ) −3 5 ( −6 ) + 7 ( −2 ) −3

    234.

    8 ( −4 ) + 5 ( −4 ) −6 8 ( −4 ) + 5 ( −4 ) −6

    235.

    −8 ( −2 ) −3 ( −9 ) −8 ( −2 ) −3 ( −9 )

    236.

    −7 ( −4 ) −5 ( −3 ) −7 ( −4 ) −5 ( −3 )

    237.

    ( −5 ) 3 ( −5 ) 3

    238.

    ( −4 ) 3 ( −4 ) 3

    239.

    ( −2 ) 6 ( −2 ) 6

    240.

    ( −3 ) 5 ( −3 ) 5

    241.

    − 4 2 − 4 2

    242.

    − 6 2 − 6 2

    243.

    −3 ( −5 ) ( 6 ) −3 ( −5 ) ( 6 )

    244.

    −4 ( −6 ) ( 3 ) −4 ( −6 ) ( 3 )

    245.

    −4 · 2 · 11 −4 · 2 · 11

    246.

    −5 · 3 · 10 −5 · 3 · 10

    247.

    ( 8 − 11 ) ( 9 − 12 ) ( 8 − 11 ) ( 9 − 12 )

    248.

    ( 6 − 11 ) ( 8 − 13 ) ( 6 − 11 ) ( 8 − 13 )

    249.

    26 − 3 ( 2 − 7 ) 26 − 3 ( 2 − 7 )

    250.

    23 − 2 ( 4 − 6 ) 23 − 2 ( 4 − 6 )

    251.

    −10 ( −4 ) ÷ ( −8 ) −10 ( −4 ) ÷ ( −8 )

    252.

    −8 ( −6 ) ÷ ( −4 ) −8 ( −6 ) ÷ ( −4 )

    253.

    65 ÷ ( −5 ) + ( −28 ) ÷ ( −7 ) 65 ÷ ( −5 ) + ( −28 ) ÷ ( −7 )

    254.

    52 ÷ ( −4 ) + ( −32 ) ÷ ( −8 ) 52 ÷ ( −4 ) + ( −32 ) ÷ ( −8 )

    255.

    9 − 2 [ 3 − 8 ( −2 ) ] 9 − 2 [ 3 − 8 ( −2 ) ]

    256.

    11 − 3 [ 7 − 4 ( −2 ) ] 11 − 3 [ 7 − 4 ( −2 ) ]

    257.

    ( −3 ) 2 −24 ÷ ( 8 − 2 ) ( −3 ) 2 −24 ÷ ( 8 − 2 )

    258.

    ( −4 ) 2 − 32 ÷ ( 12 − 4 ) ( −4 ) 2 − 32 ÷ ( 12 − 4 )

    Evaluate Variable Expressions with Integers

    In the following exercises, evaluate each expression.

    259.

    −2x+17when−2x+17when

    1. ⓐx=8x=8
    2. ⓑx=−8x=−8
    260.

    −5y+14when−5y+14when

    1. ⓐ y=9 y=9
    2. ⓑ y=−9 y=−9
    261.

    10−3mwhen 10−3mwhen

    1. ⓐ m=5 m=5
    2. ⓑ m=−5 m=−5
    262.

    18−4nwhen18−4nwhen

    1. ⓐ n=3 n=3
    2. ⓑ n=−3n=−3
    263.

    p 2 − 5 p + 5 when p = −1 p 2 − 5 p + 5 when p = −1

    264.

    q2−2q+9q2−2q+9 when q=−2q=−2

    265.

    2w2−3w+72w2−3w+7 when w=−2w=−2

    266.

    3u2−4u+53u2−4u+5 when u=−3u=−3

    267.

    6x−5y+156x−5y+15 when x=3x=3 and y=−1y=−1

    268.

    3p−2q+93p−2q+9 when p=8p=8 and q=−2q=−2

    269.

    9a−2b−89a−2b−8 when a=−6a=−6 and b=−3b=−3

    270.

    7m−4n−27m−4n−2 when m=−4m=−4 and n=−9n=−9

    Translate Word Phrases to Algebraic Expressions

    In the following exercises, translate to an algebraic expression and simplify if possible.

    271.

    The product of −3−3 and 15

    272.

    The product of −4−4 and 1616

    273.

    The quotient of −60−60 and −20−20

    274.

    The quotient of −40−40 and −20−20

    275.

    The quotient of −6−6 and the sum of aa and bb

    276.

    The quotient of −7−7 and the sum of mm and nn

    277.

    The product of −10−10 and the difference of pandqpandq

    278.

    The product of −13−13 and the difference of canddcandd

    Everyday Math

    279.

    Stock market Javier owns 300300 shares of stock in one company. On Tuesday, the stock price dropped $12$12 per share. What was the total effect on Javier’s portfolio?

    280.

    Weight loss In the first week of a diet program, eight women lost an average of 3 pounds3 pounds each. What was the total weight change for the eight women?

    Writing Exercises

    281.

    In your own words, state the rules for multiplying two integers.

    282.

    In your own words, state the rules for dividing two integers.

    283.

    Why is −24≠(−2)4?−24≠(−2)4?

    284.

    Why is −42≠(−4)2?−42≠(−4)2?

    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?

    This page titled 3.5: Multiply and Divide Integers is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.