m81278 Multiply and Divide Integers 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 25c4a027-b875-440c-bd4d-5151ed2001bd Before you get started, take this readiness quiz. Translate the quotient of 20 and 13 into an algebraic expression. If you missed this problem, review . 20÷13 Add: −5+(−5)+(−5). If you missed this problem, review . −15 Evaluaten+4whenn=−7. If you missed this problem, review . −3

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·b means add a,b times. Here, we are using the model shown in just to help us discover the pattern.

Now consider what it means to multiply 5 by −3. It means subtract 5,3 times. Looking at subtraction as taking away, it means to take away 5,3 times. But there is nothing to take away, so we start by adding neutral pairs as shown in .

In both cases, we started with 15 neutral pairs. In the case on the left, we took away 5,3 times and the result was −15. To multiply (−5)(−3), we took away −5,3 times and the result was 15. So we found that 5·3=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. The sign of the product of two numbers depends on their signs. Same signs Product •Two positives•Two negatives PositivePositive Different signs Product •Positive • negative•Negative • positive NegativeNegative Multiply each of the following: ⓐ −9·3 ⓑ −2(−5) ⓒ 4(−8) ⓓ 7·6 ⓐ –9⋅3 Multiply, noting that the signs are different and so the product is negative. –27 ⓑ –2(–5) Multiply, noting that the signs are the same and so the product is positive. 10 ⓒ 4(–8) Multiply, noting that the signs are different and so the product is negative. –32 ⓓ 7⋅6 The signs are the same, so the product is positive. 42 Multiply: ⓐ −6·8 ⓑ −4(−7) ⓒ 9(−7) ⓓ 5·12 ⓐ −48 ⓑ 28 ⓒ −63 ⓓ 60 Multiply: ⓐ −8·7 ⓑ −6(−9) ⓒ 7(−4) ⓓ 3·13 ⓐ −56 ⓑ 54 ⓒ −28 ⓓ 39 When we multiply a number by 1, the result is the same number. What happens when we multiply a number by−1? Let’s multiply a positive number and then a negative number by −1 to see what we get. −1·4−1(−3)−43−4is the opposite of43is the opposite of−3Each time we multiply a number by −1, we get its opposite. Multiplying a number by −1 gives its opposite. −1a=−a Multiply each of the following: ⓐ −1·7 ⓑ −1(−11) ⓐ The signs are different, so the product will be negative. −1⋅7 Notice that −7 is the opposite of 7. −7 ⓑ The signs are the same, so the product will be positive. −1(−11) Notice that 11 is the opposite of −11. 11 Multiply. ⓐ −1·9 ⓑ −1·(−17) ⓐ −9 ⓑ 17 Multiply. ⓐ −1·8 ⓑ −1·(−16) ⓐ −8 ⓑ 16

Division is the inverse operation of multiplication. So, 15÷3=5 because 5·3=15 In words, this expression says that 15 can be divided into 3 groups of 5 each because adding five three times gives 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=5Division 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. The sign of the quotient of two numbers depends on their signs. Same signs Quotient •Two positives•Two negatives PositivePositive Different signs Quotient •Positive & negative•Negative & positive NegativeNegative Remember, you can always check the answer to a division problem by multiplying. Divide each of the following: ⓐ −27÷3 ⓑ −100÷(−4) ⓐ –27÷3 Divide, noting that the signs are different and so the quotient is negative. –9 ⓑ –100÷(–4) Divide, noting that the signs are the same and so the quotient is positive. 25 Divide: ⓐ −42÷6 ⓑ −117÷(−3) ⓐ −7 ⓑ 39 Divide: ⓐ −63÷7 ⓑ −115÷(−5) ⓐ −9 ⓑ 23 Just as we saw with multiplication, when we divide a number by 1, the result is the same number. What happens when we divide a number by −1? Let’s divide a positive number and then a negative number by −1 to see what we get. 8÷(−1)−9÷(−1)−89−8 is the opposite of 89 is the opposite of −9When we divide a number by, −1 we get its opposite. Dividing a number by −1 gives its opposite. a÷(−1)=−a Divide each of the following: ⓐ 16÷(−1) ⓑ −20÷(−1) ⓐ 16÷(–1) The dividend, 16, is being divided by –1. –16 Dividing a number by –1 gives its opposite. Notice that the signs were different, so the result was negative. ⓑ –20÷(–1) The dividend, –20, is being divided by –1. 20 Dividing a number by –1 gives its opposite. Notice that the signs were the same, so the quotient was positive. Divide: ⓐ 6÷(−1) ⓑ −36÷(−1) ⓐ −6 ⓑ 36 Divide: ⓐ 28÷(−1) ⓑ −52÷(−1) ⓐ −28 ⓑ 52

Now we’ll simplify expressions that use all four operations–addition, subtraction, multiplication, and division–with integers. Remember to follow the order of operations. Simplify:7(−2)+4(−7)−6. We use the order of operations. Multiply first and then add and subtract from left to right. 7(−2)+4(−7)−6 Multiply first. −14+(−28)−6 Add. −42−6 Subtract. −48 Simplify: 8(−3)+5(−7)−4 −63 Simplify: 9(−3)+7(−8)−1 −84 Simplify: ⓐ (−2)4 ⓑ −24 The exponent tells how many times to multiply the base. ⓐ The exponent is 4 and the base is −2. We raise −2 to the fourth power. (−2)4 Write in expanded form. (−2)(−2)(−2)(−2) Multiply. 4(−2)(−2) Multiply. −8(−2) Multiply. 16 ⓑ The exponent is 4 and the base is 2. We raise 2 to the fourth power and then take the opposite. −24 Write in expanded form. −(2⋅2⋅2⋅2) Multiply. −(4⋅2⋅2) Multiply. −(8⋅2) Multiply. −16 Simplify: ⓐ (−3)4 ⓑ −34 ⓐ 81 ⓑ −81 Simplify: ⓐ (−7)2 ⓑ −72 ⓐ 49 ⓑ −49 Simplify:12−3(9−12). According to the order of operations, we simplify inside parentheses first. Then we will multiply and finally we will subtract. 12−3(9−12) Subtract the parentheses first. 12−3(−3) Multiply. 12−(−9) Subtract. 21 Simplify: 17−4(8−11) 29 Simplify: 16−6(7−13) 52 Simplify: 8(−9)÷(−2)3. We simplify the exponent first, then multiply and divide. 8(−9)÷(−2)3 Simplify the exponent. 8(−9)÷(−8) Multiply. −72÷(−8) Divide. 9 Simplify: 12(−9)÷(−3)3 4 Simplify: 18(−4)÷(−2)3 9 Simplify:−30÷2+(−3)(−7). First we will multiply and divide from left to right. Then we will add. −30÷2+(−3)(−7) Divide. −15+(−3)(−7) Multiply. −15+21 Add. 6 Simplify: −27÷3+(−5)(−6) 21 Simplify: −32÷4+(−2)(−7) 6

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. Evaluate2x2−3x+8whenx=−4. 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·−42−3·−4+8. Evaluate: 3x2−2x+6whenx=−3 39 Evaluate: 4x2−x−5whenx=−2 13 Evaluate3x+4y−6whenx=−1andy=2. Substitute x=−1 and y=2. Multiply. Simplify. Evaluate: 7x+6y−12whenx=−2andy=3 −8 Evaluate: 8x−6y+13whenx=−3andy=−5 19

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. The word product tells us to multiply. the product of −2 and 14 Translate. (−2)(14) Simplify. −28 Translate to an algebraic expression and simplify if possible: the product of −5 and 12 −5 (12) = −60 Translate to an algebraic expression and simplify if possible: the product of 8 and −13 8 (−13) = −104 Translate to an algebraic expression and simplify if possible: the quotient of −56 and −7. The word quotient tells us to divide. the quotient of −56 and −7 Translate. −56÷(−7) Simplify. 8 Translate to an algebraic expression and simplify if possible: the quotient of −63 and −9 −63 ÷ −9 = 7 Translate to an algebraic expression and simplify if possible: the quotient of −72 and −9 −72 ÷ −9 = 8 Multiplying Integers Using Color Counters Multiplying Integers Using Color Counters With Neutral Pairs Multiplying Integers Basics Dividing Integers Basics Ex. Dividing Integers Multiplying and Dividing Signed Numbers

Multiplication of Signed Numbers To determine the sign of the product of two signed numbers: Same Signs Product Two positivesTwo negatives PositivePositive Different Signs Product Positive • negativeNegative • positive NegativeNegative Division of Signed Numbers To determine the sign of the quotient of two signed numbers: Same Signs Quotient Two positivesTwo negatives PositivePositive Different Signs Quotient Positive • negativeNegative • Positive NegativeNegative 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÷(−1)=−a

Multiply Integers In the following exercises, multiply each pair of integers. −4·8 −32 −3·9 −5(7) −35 −8(6) −18(−2) 36 −10(−6) 9(−7) −63 13(−5) −1·6 −6 −1·3 −1(−14) 14 −1(−19) Divide Integers In the following exercises, divide. −24÷6 −4 −28÷7 56÷(−7) −8 35÷(−7) −52÷(−4) 13 −84÷(−6) −180÷15 −12 −192÷12 49÷(−1) −49 62÷(−1) Simplify Expressions with Integers In the following exercises, simplify each expression. 5(−6)+7(−2)−3 −47 8(−4)+5(−4)−6 −8(−2)−3(−9) 43 −7(−4)−5(−3) (−5)3 −125 (−4)3 (−2)6 64 (−3)5 −42 −16 −62 −3(−5)(6) 90 −4(−6)(3) −4·2·11 −88 −5·3·10 (8−11)(9−12) 9 (6−11)(8−13) 26−3(2−7) 41 23−2(4−6) −10(−4)÷(−8) −5 −8(−6)÷(−4) 65÷(−5)+(−28)÷(−7) −9 52÷(−4)+(−32)÷(−8) 9−2[3−8(−2)] −29 11−3[7−4(−2)] (−3)2−24÷(8−2) 5 (−4)2−32÷(12−4) Evaluate Variable Expressions with Integers In the following exercises, evaluate each expression. −2x+17when ⓐ x=8 ⓑ x=−8 ⓐ 1 ⓑ 33 −5y+14when ⓐ y=9 ⓑ y=−9 10−3mwhen ⓐ m=5 ⓑ m=−5 ⓐ −5 ⓑ 25 18−4nwhen ⓐ n=3 ⓑ n=−3 p2−5p+5whenp=−1 11 q2−2q+9 when q=−2 2w2−3w+7 when w=−2 21 3u2−4u+5 when u=−3 6x−5y+15 when x=3 and y=−1 38 3p−2q+9 when p=8 and q=−2 9a−2b−8 when a=−6 and b=−3 −56 7m−4n−2 when m=−4 and 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 −6a+b The quotient of −7 and the sum of m and n The product of −10 and the difference of pandq −10 (p − q) The product of −13 and the difference of candd

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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 −24≠(−2)4? Sample answer: In the first expression the base is positive and after you raise it to the power you should take the opposite. Then in the second expression the base is negative so you simply raise it to the power. Why is −42≠(−4)2?

ⓐ 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?