3.4 Multiply and Divide Integers

Learning objectives

Learning objectives

• Multiply integers

• Divide integers

• Simplify expressions with integers

• Evaluate variable expressions with integers

• Translate word phrases to algebraic expressions

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 $a\cdot{}b$ means add $a,\,b$ times. Here, we are using the model shown in Figure 3.19 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 Figure 3.20.

In both cases, we started with $\mathrm{15}$ neutral pairs. In the case on the left, we took away $5,3$ times and the result was $-\mathrm{15}.$ To multiply $(-5)(-3),$ we took away $-5,3$ times and the result was $\mathrm{15}.$ So we found that

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.

Example 3.47

Solution

1. Choose the sign rule. A product is negative when its factors have unlike signs and positive when its factors have like signs.
2. Multiply ⓐ. The magnitudes $9$ and $3$ give $27$, and unlike signs make $-9\cdot3=-27$.
3. Multiply ⓑ. The magnitudes $2$ and $5$ give $10$, and two negative signs make $-2(-5)=10$.
4. Multiply ⓒ. The magnitudes $4$ and $8$ give $32$, and unlike signs make $4(-8)=-32$.
5. Multiply ⓓ. The positive factors $7$ and $6$ give $42$, so $7\cdot6=42$.
6. Check the sign pattern. The products have signs negative, positive, negative, and positive in the order listed.

Takeaway: Products with unlike signs are negative and products with like signs are positive, giving $-27$, $10$, $-32$, and $42$ in order.

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.

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

Example 3.48

Solution

1. Use the rule for $-1$. Multiplying any integer by $-1$ gives its opposite.
2. Multiply ⓐ. In $-1\cdot7$, the signs differ, so the product is $-7$.
3. Check ⓐ. The number $-7$ is indeed the opposite of the original factor $7$.
4. Multiply ⓑ. In $-1(-11)$, the signs are the same, so the product is $11$.
5. Check ⓑ. The number $11$ is the opposite of the original factor $-11$.

Takeaway: Multiplying by $-1$ gives the opposite, so $-1\cdot{}7=-7$ while $-1(-11)=11$.

Divide Integers

Division is the inverse operation of multiplication. So, $15\mathrm{÷}3=5$ because $5\cdot{}3=15$ In words, this expression says that $\mathrm{15}$ can be divided into $3$ groups of $5$ each because adding five three times gives $\mathrm{15}.$ If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.

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.

Remember, you can always check the answer to a division problem by multiplying.

Example 3.49

Solution

1. Choose the quotient sign rule. A quotient is negative for unlike signs and positive for like signs.
2. Divide ⓐ. The magnitudes give $27\div3=9$, and unlike signs make $-27\div3=-9$.
3. Divide ⓑ. The magnitudes give $100\div4=25$, and two negative signs make $-100\div(-4)=25$.
4. Check the divisions. Multiplying $-9$ by $3$ returns $-27$, while $25(-4)=-100$.
5. Confirm the signs. The first quotient is negative and the second is positive, exactly as the rule predicts.

Takeaway: A quotient has a negative sign when its dividend and divisor have unlike signs, so $-27\div{}3=-9$ and $-100\div{}(-4)=25$.

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.

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

Example 3.50

Solution

1. Identify the divisor. Each division uses $-1$, so each quotient is the opposite of its dividend.
2. Divide ⓐ. The opposite of $16$ is $-16$, so $16\div(-1)=-16$.
3. Check ⓐ. Multiplying $-16$ by $-1$ returns $16$, confirming the quotient.
4. Divide ⓑ. The opposite of $-20$ is $20$, so $-20\div(-1)=20$.
5. Check ⓑ. Multiplying $20$ by $-1$ returns $-20$, confirming the quotient.

Takeaway: Dividing by $-1$ produces the opposite of the dividend, so $16\div{}(-1)=-16$ and $-20\div{}(-1)=20$.

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

Solution

1. Apply the order of operations. Multiply each parenthesized factor before adding and subtracting the resulting terms.
2. Multiply the first factor. Compute $7(-2)=-14$, giving $-14+4(-7)-6$.
3. Multiply the second factor. Compute $4(-7)=-28$, giving $-14+(-28)-6$.
4. Combine the negative terms. Add $-14$ and $-28$ to obtain $-42$, so the expression becomes $-42-6$.
5. Subtract and check. Compute $-42-6=-48$, and both products were completed before the additions.

Takeaway: Apply multiplication before addition and subtraction, so $7(-2)+4(-7)-6=-14-28-6=-48$.

Example 3.52

Solution

1. Identify the base. Parentheses make $-2$ the whole base in $(-2)^4$, but only $2$ is the base in $-2^4$.
2. Expand ⓐ. Write $(-2)^4$ as four factors of $-2$: $(-2)(-2)(-2)(-2)$.
3. Evaluate ⓐ. Combine the factors to obtain $4(-2)(-2)=(-8)(-2)=16$.
4. Expand ⓑ. Write $-2^4$ as $-(2\cdot2\cdot2\cdot2)$, with the negative sign outside the power.
5. Evaluate ⓑ. The four positive factors give $16$, and the outside negative sign makes the result $-16$.
6. Compare the two powers. Parentheses give $(-2)^4=16$, while $-2^4=-(2^4)=-16$.

Takeaway: Parentheses make $-2$ the entire base, so ${(-2)}^4=16$, while $-2^4=-(2^4)=-16$.

Example 3.53

Solution

1. Simplify the parentheses first. In $12-3(9-12)$, compute $9-12=-3$ before multiplying.
2. Rewrite the expression. Substitution gives $12-3(-3)$.
3. Multiply. Compute $3(-3)=-9$, so the expression becomes $12-(-9)$.
4. Subtract. Subtracting a negative adds $9$, giving $12+9=21$.
5. Check the order. The grouped subtraction was completed before the multiplication and final subtraction.

Takeaway: Simplify the parentheses before multiplying and subtracting, so $12-3(9-12)=12-3(-3)=21$.

Example 3.54

Solution

1. Evaluate the exponent first. In $8(-9)\div{}(-2)^3$, compute the power before multiplying or dividing.
2. Simplify the power. Three negative factors give $(-2)^3=-8$, so the expression becomes $8(-9)\div(-8)$.
3. Multiply the numerator terms. Compute $8(-9)=-72$, giving $-72\div(-8)$.
4. Divide the signed values. Two negative signs make the quotient positive, and $72\div8=9$.
5. Check the result. Multiplying $9$ by $-8$ returns $-72$, confirming the quotient.

Takeaway: Compute the exponent before multiplying and dividing, so $8(-9)\div{}(-2)^3=-72\div{}(-8)=9$.

Example 3.55

Solution

1. Complete multiplication and division first. In $-30\div2+(-3)(-7)$, simplify both operations before adding.
2. Divide the first term. Compute $-30\div2=-15$, giving $-15+(-3)(-7)$.
3. Multiply the second term. Since the two negatives multiply to a positive, compute $(-3)(-7)=21$.
4. Add the results. The expression is now $-15+21$.
5. Simplify the sum. Compute $-15+21=6$, which agrees with the sign and magnitude check.

Takeaway: Multiply and divide before adding, so $-30\div{}2+(-3)(-7)=-15+21=6$.

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.

Example 3.56

Solution

1. Substitute the value. Replace $x$ with $-4$ in $2x^2-3x+8$ to obtain $2(-4)^2-3(-4)+8$, as shown in the substitution figure.
2. Evaluate the power. Compute $(-4)^2=16$, giving $2(16)-3(-4)+8$, matching the next figure.
3. Multiply. Compute $2(16)=32$ and $-3(-4)=12$, so the expression becomes $32+12+8$.
4. Add. Compute $32+12=44$, matching the figure showing $44+8$.
5. State the result. Adding $8$ gives $52$, as shown in the final figure, so the expression equals $52$ when $x=-4$.

Takeaway: Use parentheses when substituting a negative value so that the power belongs to the value and not to the coefficient.

Example 3.57

Solution

1. Substitute both values. Replace $x$ with $-1$ and $y$ with $2$ in $3x+4y-6$, matching the first two figures.
2. Multiply each term. Compute $3(-1)=-3$ and $4(2)=8$, giving $-3+8-6$, as shown in the next figure.
3. Add the first two terms. Compute $-3+8=5$, so the expression becomes $5-6$.
4. Subtract the last term. Compute $5-6=-1$, matching the final figure.
5. State the result. The expression $3x+4y-6$ equals $-1$ when $x=-1$ and $y=2$.

Takeaway: Substitute every variable and complete each multiplication before adding or subtracting the resulting terms.

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

Solution

1. Interpret the key word. The phrase the product of identifies multiplication of the two stated numbers.
2. Write the expression. Place $-2$ and $14$ as factors to obtain $(-2)(14)$.
3. Multiply the magnitudes. Compute $2\cdot14=28$ to find the numerical magnitude of the product.
4. Apply the sign rule. The factor $-2$ makes the product negative, so $(-2)(14)=-28$.

Takeaway: The word product means multiplication, so the product of $-2$ and $14$ is $(-2)(14)=-28$.

Example 3.59

Solution

1. Interpret the key word. The phrase the quotient of identifies division, with the first number as the dividend.
2. Write the expression. Place $-56$ before the division sign and $-7$ after it to get $-56\div(-7)$.
3. Divide the magnitudes. Compute $56\div7=8$ to find the numerical magnitude of the quotient.
4. Apply the sign rule. Both dividend and divisor are negative, so the quotient is positive: $-56\div(-7)=8$.

Takeaway: The word quotient means division, so the quotient of $-56$ and $-7$ is $-56\div{}(-7)=8$.

Section 3.4 Exercises

Section 3.4 Exercises

Multiply Integers

In the following exercises, multiply each pair of integers.

Divide Integers

In the following exercises, divide.

Simplify Expressions with Integers

In the following exercises, simplify each expression.

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Translate Word Phrases to Algebraic Expressions

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

Everyday Math

Writing Exercises

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?

1. Multiply:

1. ⓐ $\,-6\cdot{}8\,$

2. ⓑ $\,-4(-7)\,$

3. ⓒ $\,9(-7)\,$

4. ⓓ $\,5\cdot{}12$

2. Multiply:

1. ⓐ $\,-8\cdot{}7\,$

2. ⓑ $\,-6(-9)\,$

3. ⓒ $\,7(-4)\,$

4. ⓓ $\,3\cdot{}13$

3. Multiply.

1. ⓐ $\,-1\cdot{}9\,$

2. ⓑ $\,-1\cdot(-17)$

1. Multiply.

1. ⓐ $\,-1\cdot{}8\,$

2. ⓑ $\,-1\cdot(-16)$

2. Divide:

1. ⓐ $\,-42\mathrm{÷}6\,$

2. ⓑ $\,-117\mathrm{÷}(-3)$

3. Divide:

1. ⓐ $\,-63\mathrm{÷}7\,$

2. ⓑ $\,-115\mathrm{÷}(-5)$

1. Divide:

1. ⓐ $\,6\mathrm{÷}(-1)\,$

2. ⓑ $\,-36\mathrm{÷}(-1)$

2. Divide:

1. ⓐ $\,28\mathrm{÷}(-1)\,$

2. ⓑ $\,-52\mathrm{÷}(-1)$

3. Simplify:

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

1. Simplify:

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

2. Simplify:

1. ⓐ $\,{(-3)}^{4}\,$

2. ⓑ $\,{-3}^{4}$

3. Simplify:

1. ⓐ $\,{(-7)}^{2}\,$

2. ⓑ $\,-{7}^{2}$

1. Evaluate:

$7x+6y-12\,\mathrm{when}\,x=-2\,\mathrm{and}\,y=3$

2. Translate to an algebraic expression and simplify if possible:

$\mathrm{the product of -5 and 12}$

3. Translate to an algebraic expression and simplify if possible:

$\mathrm{the product of 8 and -13}$

1. Translate to an algebraic expression and simplify if possible:

$\mathrm{the quotient of -63 and -9}$

2. Translate to an algebraic expression and simplify if possible:

$\mathrm{the quotient of -72 and -9}$

3. $-4\cdot{}8$

1. $-5(7)$

2. $9(-7)$

3. $-1\cdot{}6$

1. $-24\mathrm{÷}6$

2. $56\mathrm{÷}(-7)$

3. $-180\mathrm{÷}15$

1. $49\mathrm{÷}(-1)$

2. $5(-6)+7(-2)-3$

3. ${(-5)}^{3}$

1. $-{4}^{2}$

2. $-4\cdot{}2\cdot{}11$

3. $-10(-4)\mathrm{÷}(-8)$

1. $65\mathrm{÷}(-5)+(-28)\mathrm{÷}(-7)$

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

3. $-2x+17\,\mathrm{when}\,$

1. ⓐ $\,x=8\,$

2. ⓑ $\,x=-8$

1. $10-3m\,\mathrm{when}\,$

1. ⓐ $\,m=5\,$

2. ⓑ $\,m=-5$

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

3. The product of $-3$ and 15

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

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

3. The product of $-10$ and the difference of $p\,\mathrm{and}\,q$

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

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

3. Why is ${-2}^{4}\ne{(-2)}^{4}?$