# 3.4 Multiply and divide integers  (Page 3/3)

 Page 3 / 3

Simplify:

$18\left(-4\right)÷{\left(-2\right)}^{3}$

9

$\text{Simplify:}\phantom{\rule{0.2em}{0ex}}-30÷2+\left(-3\right)\left(-7\right).$

## Solution

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

 $-30÷2+\left(-3\right)\left(-7\right)$ Divide. $-15+\left(-3\right)\left(-7\right)$ Multiply. $-15+21$ Add. $\text{6}$

Simplify:

$-27÷3+\left(-5\right)\left(-6\right)$

21

Simplify:

$-32÷4+\left(-2\right)\left(-7\right)$

6

## 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.

$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}2{x}^{2}-3x+8\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-4.$

## Solution   Simplify exponents. Multiply. Subtract. Add. Keep in mind that when we substitute $-4$ for $x,$ we use parentheses to show the multiplication. Without parentheses, it would look like $2·{-4}^{2}-3·-4+8.$

Evaluate:

$3{x}^{2}-2x+6\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-3$

39

Evaluate:

$4{x}^{2}-x-5\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-2$

13

$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}3x+4y-6\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-1\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=2.$

## Solution Substitute $x=-1$ and $y=2$ . Multiply. Simplify. Evaluate:

$7x+6y-12\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-2\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=3$

−8

Evaluate:

$8x-6y+13\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-3\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=-5$

19

## 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 .

Translate to an algebraic expression and simplify if possible: the product of $-2$ and $14.$

## Solution

The word product tells us to multiply.

 the product of $-2$ and $14$ Translate. $\left(-2\right)\left(14\right)$ Simplify. $-28$

Translate to an algebraic expression and simplify if possible:

$\text{the product of −5 and 12}$

−5 (12) = −60

Translate to an algebraic expression and simplify if possible:

$\text{the product of 8 and −13}$

8 (−13) = −104

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

## Solution

The word quotient tells us to divide.

 the quotient of −56 and −7 Translate. $-56÷\left(-7\right)$ Simplify. $8$

Translate to an algebraic expression and simplify if possible:

$\text{the quotient of −63 and −9}$

−63 ÷ −9 = 7

Translate to an algebraic expression and simplify if possible:

$\text{the quotient of −72 and −9}$

−72 ÷ −9 = 8

## Key concepts

• Multiplication and Division of Signed Numbers
Same signs Product
•Two positives
•Two negatives
Positive
Positive
Different signs Product
•Positive • negative
•Negative • positive
Negative
Negative
• Multiplication by $-1$
• Multiplying a number by $-1$ gives its opposite: $-1a=-a$
• Division by $-1$
• Dividing a number by $-1$ gives its opposite: $a÷\left(-1\right)=\mathit{\text{−a}}$

## Practice makes perfect

Multiply Integers

In the following exercises, multiply each pair of integers.

$-4·8$

−32

$-3·9$

$-5\left(7\right)$

−35

$-8\left(6\right)$

$-18\left(-2\right)$

36

$-10\left(-6\right)$

$9\left(-7\right)$

−63

$13\left(-5\right)$

$-1·6$

−6

$-1·3$

$-1\left(-14\right)$

14

$-1\left(-19\right)$

Divide Integers

In the following exercises, divide.

$-24÷6$

−4

$-28÷7$

$56÷\left(-7\right)$

−8

$35÷\left(-7\right)$

$-52÷\left(-4\right)$

13

$-84÷\left(-6\right)$

$-180÷15$

−12

$-192÷12$

$49÷\left(-1\right)$

−49

$62÷\left(-1\right)$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

$5\left(-6\right)+7\left(-2\right)-3$

−47

$8\left(-4\right)+5\left(-4\right)-6$

$-8\left(-2\right)-3\left(-9\right)$

43

$-7\left(-4\right)-5\left(-3\right)$

${\left(-5\right)}^{3}$

−125

${\left(-4\right)}^{3}$

${\left(-2\right)}^{6}$

64

${\left(-3\right)}^{5}$

$-{4}^{2}$

−16

$-{6}^{2}$

$-3\left(-5\right)\left(6\right)$

90

$-4\left(-6\right)\left(3\right)$

$-4·2·11$

−88

$-5·3·10$

$\left(8-11\right)\left(9-12\right)$

9

$\left(6-11\right)\left(8-13\right)$

$26-3\left(2-7\right)$

41

$23-2\left(4-6\right)$

$-10\left(-4\right)÷\left(-8\right)$

−5

$-8\left(-6\right)÷\left(-4\right)$

$65÷\left(-5\right)+\left(-28\right)÷\left(-7\right)$

−9

$52÷\left(-4\right)+\left(-32\right)÷\left(-8\right)$

$9-2\left[3-8\left(-2\right)\right]$

−29

$11-3\left[7-4\left(-2\right)\right]$

${\left(-3\right)}^{2}-24÷\left(8-2\right)$

5

${\left(-4\right)}^{2}-32÷\left(12-4\right)$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

$-2x+17\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$

1. $\phantom{\rule{0.2em}{0ex}}x=8\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}x=-8$

1. 1
2. 33

$-5y+14\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$

1. $\phantom{\rule{0.2em}{0ex}}y=9\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}y=-9$

$10-3m\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$

1. $\phantom{\rule{0.2em}{0ex}}m=5\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}m=-5$

1. −5
2. 25

$18-4n\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$

1. $\phantom{\rule{0.2em}{0ex}}n=3\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}n=-3$

${p}^{2}-5p+5\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=-1$

8

${q}^{2}-2q+9\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}q=-2$

$2{w}^{2}-3w+7\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}w=-2$

21

$3{u}^{2}-4u+5\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}u=-3$

$6x-5y+15\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=3\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=-1$

38

$3p-2q+9\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=8\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}q=-2$

$9a-2b-8\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}a=-6\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b=-3$

−56

$7m-4n-2\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}m=-4\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n=-9$

Translate Word Phrases to Algebraic Expressions

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

The product of $-3$ and 15

−3·15 = −45

The product of $-4$ and $16$

The quotient of $-60$ and $-20$

−60 ÷ (−20) = 3

The quotient of $-40$ and $-20$

The quotient of $-6$ and the sum of $a$ and $b$

$\frac{-6}{a+b}$

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

The product of $-10$ and the difference of $p\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}q$

−10 ( p q )

The product of $-13$ and the difference of $c\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d$

## Everyday math

Stock market Javier owns $300$ shares of stock in one company. On Tuesday, the stock price dropped $\text{12}$ per share. What was the total effect on Javier’s portfolio?

−$3,600 Weight loss In the first week of a diet program, eight women lost an average of $\text{3 pounds}$ each. What was the total weight change for the eight women? ## Writing exercises In your own words, state the rules for multiplying two integers. Sample answer: Multiplying two integers with the same sign results in a positive product. Multiplying two integers with different signs results in a negative product. In your own words, state the rules for dividing two integers. Why is ${-2}^{4}\ne {\left(-2\right)}^{4}?$ Sample answer: In one expression the base is positive and then we take the opposite, but in the other the base is negative. Why is ${-4}^{2}\ne {\left(-4\right)}^{2}?$ ## 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? 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