Contents

Multiply and Divide Integers

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 3.4

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.
Add: $−5 + (−5) + (−5) . −5 + (−5) + (−5) .$
If you missed this problem, review Example 3.21.
$Evaluate n + 4 when n = −7 . Evaluate$
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 $a \cdot b a \cdot b$ means add $a, b a,$ times. Here, we are using the model shown in Figure 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 $55$ by $−3 . −3 .$ It means subtract $5, 3 5, 3$ times. Looking at subtraction as taking away, it means to take away $5, 3 5, 3$ times. But there is nothing to take away, so we start by adding neutral pairs as shown in Figure 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 $5, 3 5, 3$ 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

\begin{matrix} 5 \cdot 3 = 15 & −5 (3) = −15 \\ 5 (−3) = −15 & (−5) (−3) = 15 \end{matrix} \begin{matrix} 5 \cdot 3 = 15 & −5 (3) = −15 \\ 5 (−3) = −15 & (−5) (−3) = 15 \end{matrix}

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:

ⓐ $−9 \cdot 3$
ⓑ $−2 (−5)$
ⓒ $4 (−8)$
ⓓ $7 \cdot 6$

Show solution

Solution

ⓐ
	$–9 \cdot 3 –9 \cdot 3$
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

ⓓ
	$7 \cdot 6 7 \cdot 6$
The signs are the same, so the product is positive.	$4242$

Table 3.8

Try It 3.93

Multiply:

ⓐ $−6 \cdot 8$
ⓑ $−4 (−7)$
ⓒ $9 (−7)$
ⓓ $5 \cdot 12$

Try It 3.94

Multiply:

ⓐ $−8 \cdot 7$
ⓑ $−6 (−9)$
ⓒ $7 (−4)$
ⓓ $3 \cdot 13$

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.

\begin{matrix} −1 \cdot 4 & −1 (−3) \\ −4 & 3 \\ −4 is the opposite of 4 & 3 is the opposite of −3 \end{matrix} \begin{matrix} −1 \cdot 4 & −1 (−3) \\ −4 & 3 \\ −4 & 3 \end{matrix}

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.

−1 a = - a −1 a = - a

Example 3.48

Multiply each of the following:

ⓐ $−1 \cdot 7$
ⓑ $−1 (−11)$

Show solution

Solution

ⓐ
The signs are different, so the product will be negative.	$−1 \cdot 7 −1 \cdot 7$
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.

ⓐ $−1 \cdot 9$
ⓑ $−1 \cdot (−17)$

Try It 3.96

Multiply.

ⓐ $−1 \cdot 8$
ⓑ $−1 \cdot (−16)$

Divide Integers

Division is the inverse operation of multiplication. So, $15 \div 3 = 5 15 \div 3 = 5$ because $5 \cdot 3 = 15 5 \cdot 3 = 15$ 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.

\begin{matrix} 5 \cdot 3 = 15 so 15 \div 3 = 5 & −5 (3) = −15 so −15 \div 3 = −5 \\ (−5) (−3) = 15 so 15 \div (−3) = −5 & 5 (−3) = −15 so −15 \div −3 = 5 \end{matrix} \begin{matrix} 5 \cdot 3 = 15 & −5 (3) = −15 \\ (−5) (−3) = 15 & 5 (−3) = −15 \end{matrix}

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:

ⓐ $−27 \div 3$
ⓑ $−100 \div (−4)$

Show solution

Solution

ⓐ
	$–27 \div 3 –27 \div 3$
Divide, noting that the signs are different and so the quotient is negative.	$–9 –9$

Table 3.9

ⓑ
	$–100 \div (–4) –100 \div (–4)$
Divide, noting that the signs are the same and so the quotient is positive.	$2525$

Table 3.10

Try It 3.97

Divide:

ⓐ $−42 \div 6$
ⓑ $−117 \div (−3)$

Try It 3.98

Divide:

ⓐ $−63 \div 7$
ⓑ $−115 \div (−5)$

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.

\begin{matrix} 8 \div (−1) & −9 \div (−1) \\ −8 & 9 \\ −8 is the opposite of 8 & 9 is the opposite of −9 \end{matrix} \begin{matrix} 8 \div (−1) & −9 \div (−1) \\ −8 & 9 \\ −8 is the opposite of 8 & 9 is the opposite of −9 \end{matrix}

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 \div (−1) = −a a \div (−1) = −a

Example 3.50

Divide each of the following:

ⓐ $16 \div (−1)$
ⓑ $−20 \div (−1)$

Show solution

Solution

ⓐ
	$16 \div (–1) 16 \div (–1)$
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

ⓑ
	$–20 \div (–1) –20 \div (–1)$
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:

ⓐ $6 \div (−1)$
ⓑ $−36 \div (−1)$

Try It 3.100

Divide:

ⓐ $28 \div (−1)$
ⓑ $−52 \div (−1)$

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

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

Show solution

Solution

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

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

Try It 3.101

Simplify:

$8 (−3) + 5 (−7) −4 8 (−3) + 5 (−7) −4$

Try It 3.102

Simplify:

$9 (−3) + 7 (−8) - 1 9 (−3) + 7 (−8) - 1$

Example 3.52

Simplify:

ⓐ ${(−2)}^{4}$
ⓑ ${−2}^{4}$

Show solution

Solution

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.

	$- 2^{4} - 2^{4}$
Write in expanded form.	$- (2 \cdot 2 \cdot 2 \cdot 2) - (2 \cdot 2 \cdot 2 \cdot 2)$
Multiply.	$- (4 \cdot 2 \cdot 2) - (4 \cdot 2 \cdot 2)$
Multiply.	$- (8 \cdot 2) - (8 \cdot 2)$
Multiply.	$- 16 - 16$

Try It 3.103

Simplify:

ⓐ ${(−3)}^{4}$
ⓑ ${−3}^{4}$

Try It 3.104

Simplify:

ⓐ ${(−7)}^{2}$
ⓑ $- 7^{2}$

Example 3.53

$Simplify: 12 - 3 (9 - 12) . Simplify:$

Show solution

Solution

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: $8 (−9) \div {(−2)}^{3} . 8 (−9) \div {(−2)}^{3} .$

Show solution

Solution

We simplify the exponent first, then multiply and divide.

	$8 (−9) \div {(−2)}^{3} 8 (−9) \div {(−2)}^{3}$
Simplify the exponent.	$8 (−9) \div (−8) 8 (−9) \div (−8)$
Multiply.	$−72 \div (−8) −72 \div (−8)$
Divide.	$99$

Try It 3.107

Simplify:

$12 (−9) \div {(−3)}^{3} 12 (−9) \div {(−3)}^{3}$

Try It 3.108

Simplify:

$18 (−4) \div {(−2)}^{3} 18 (−4) \div {(−2)}^{3}$

Example 3.55

$Simplify: −30 \div 2 + (−3) (−7) . Simplify:$

Show solution

Solution

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

	$−30 \div 2 + (−3) (−7) −30 \div 2 + (−3) (−7)$
Divide.	$−15 + (−3) (−7) −15 + (−3) (−7)$
Multiply.	$−15 + 21 −15 + 21$
Add.	$66$

Try It 3.109

Simplify:

$−27 \div 3 + (−5) (−6) −27 \div 3 + (−5) (−6)$

Try It 3.110

Simplify:

$−32 \div 4 + (−2) (−7) −32 \div 4 + (−2) (−7)$

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

$Evaluate 2 x^{2} - 3 x + 8 when x = −4 . Evaluate$

Show solution

Solution



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 $2 \cdot {−4}^{2} - 3 \cdot −4 + 8 . 2 \cdot {−4}^{2} - 3 \cdot −4 + 8 .$

Try It 3.111

Evaluate:

$3 x^{2} - 2 x + 6 when x = −3 3 x^{2} - 2 x + 6$

Try It 3.112

Evaluate:

$4 x^{2} - x - 5 when x = −2 4 x^{2} - x - 5$

Example 3.57

$Evaluate 3 x + 4 y - 6 when x = −1 and y = 2 . Evaluate$

Show solution

Solution


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

Try It 3.113

Evaluate:

$7 x + 6 y - 12 when x = −2 and y = 3 7 x + 6 y - 12$

Try It 3.114

Evaluate:

$8 x - 6 y + 13 when x = −3 and y = −5 8 x - 6 y + 13$

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 $14 . 14 .$

Show solution

Solution

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:

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

Try It 3.116

Translate to an algebraic expression and simplify if possible:

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

Example 3.59

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

Show solution

Solution

The word quotient tells us to divide.

	the quotient of −56 and −7
Translate.	$−56 \div (−7) −56 \div (−7)$
Simplify.	$88$

Try It 3.117

Translate to an algebraic expression and simplify if possible:

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

Try It 3.118

Translate to an algebraic expression and simplify if possible:

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

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Practice Makes Perfect

Multiply Integers

In the following exercises, multiply each pair of integers.

211.

$−4 \cdot 8 −4 \cdot 8$

212.

$−3 \cdot 9 −3 \cdot 9$

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.

$−1 \cdot 6 −1 \cdot 6$

220.

$−1 \cdot 3 −1 \cdot 3$

221.

$−1 (−14) −1 (−14)$

222.

$−1 (−19) −1 (−19)$

Divide Integers

In the following exercises, divide.

223.

$−24 \div 6 −24 \div 6$

224.

$−28 \div 7 −28 \div 7$

225.

$56 \div (−7) 56 \div (−7)$

226.

$35 \div (−7) 35 \div (−7)$

227.

$−52 \div (−4) −52 \div (−4)$

228.

$−84 \div (−6) −84 \div (−6)$

229.

$−180 \div 15 −180 \div 15$

230.

$−192 \div 12 −192 \div 12$

231.

$49 \div (−1) 49 \div (−1)$

232.

$62 \div (−1) 62 \div (−1)$

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.

$−4 \cdot 2 \cdot 11 −4 \cdot 2 \cdot 11$

246.

$−5 \cdot 3 \cdot 10 −5 \cdot 3 \cdot 10$

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.

$−10 (−4) \div (−8) −10 (−4) \div (−8)$

252.

$−8 (−6) \div (−4) −8 (−6) \div (−4)$

253.

$65 \div (−5) + (−28) \div (−7) 65 \div (−5) + (−28) \div (−7)$

254.

$52 \div (−4) + (−32) \div (−8) 52 \div (−4) + (−32) \div (−8)$

255.

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

256.

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

257.

${(−3)}^{2} −24 \div (8 - 2) {(−3)}^{2} −24 \div (8 - 2)$

258.

${(−4)}^{2} - 32 \div (12 - 4) {(−4)}^{2} - 32 \div (12 - 4)$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

259.

$−2 x + 17 when −2 x + 17$

ⓐ $x = 8$
ⓑ $x = −8$

260.

$−5 y + 14 when −5 y + 14$

ⓐ $y = 9$
ⓑ $y = −9$

261.

$10 - 3 m when 10 - 3 m$

ⓐ $m = 5$
ⓑ $m = −5$

262.

$18 - 4 n when 18 - 4 n$

ⓐ $n = 3$
ⓑ $n = −3$

263.

$p^{2} - 5 p + 5 when p = −1 p^{2} - 5 p + 5$

264.

$q^{2} - 2 q + 9 q^{2} - 2 q + 9$ when $q = −2 q = −2$

265.

$2 w^{2} - 3 w + 7 2 w^{2} - 3 w + 7$ when $w = −2 w = −2$

266.

$3 u^{2} - 4 u + 5 3 u^{2} - 4 u + 5$ when $u = −3 u = −3$

267.

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

268.

$3 p - 2 q + 9 3 p - 2 q + 9$ when $p = 8 p = 8$ and $q = −2 q = −2$

269.

$9 a - 2 b - 8 9 a - 2 b - 8$ when $a = −6 a = −6$ and $b = −3 b = −3$

270.

$7 m - 4 n - 2 7 m - 4 n - 2$ when $m = −4 m = −4$ and $n = −9 n = −9$

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 $a a$ and $b b$

276.

The quotient of $−7 −7$ and the sum of $m m$ and $n n$

277.

The product of $−10 −10$ and the difference of $p and q p$

278.

The product of $−13 −13$ and the difference of $c and d c$

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 $3 pounds 3 pounds$ 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 ${−2}^{4} \neq {(−2)}^{4} ? {−2}^{4} \neq {(−2)}^{4} ?$

284.

Why is ${−4}^{2} \neq {(−4)}^{2} ? {−4}^{2} \neq {(−4)}^{2} ?$

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?

Learning Objectives

Be Prepared 3.4

Multiply Integers

Multiplication of Signed Numbers

Example 3.47

Solution

Try It 3.93

Try It 3.94

Multiplication by −1 −1

Example 3.48

Solution

Try It 3.95

Try It 3.96

Divide Integers

Division of Signed Numbers

Example 3.49

Solution

Try It 3.97

Try It 3.98

Division by −1 −1

Example 3.50

Solution

Try It 3.99

Try It 3.100

Simplify Expressions with Integers

Example 3.51

Solution

Try It 3.101

Try It 3.102

Example 3.52

Solution

Try It 3.103

Try It 3.104

Example 3.53

Solution

Try It 3.105

Try It 3.106

Example 3.54

Solution

Try It 3.107

Try It 3.108

Example 3.55

Solution

Try It 3.109

Try It 3.110

Evaluate Variable Expressions with Integers

Example 3.56

Solution

Try It 3.111

Try It 3.112

Example 3.57

Solution

Try It 3.113

Try It 3.114

Translate Word Phrases to Algebraic Expressions

Example 3.58

Solution

Try It 3.115

Try It 3.116

Example 3.59

Solution

Try It 3.117

Try It 3.118

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Practice Makes Perfect

Everyday Math

Writing Exercises

Self Check

Multiplication by $−1 −1$

Division by $−1 −1$