UNIT TWO: Prealgebra in a Technical World
2.4 Multiplying Integers
SWBAT 1. Multiply integers.
2. Use multiplication properties of 0 and -1.
3. Simplify integer expressions containing multiplication.
4. Simplify integer expressions containing exponents.
5. Solve applied problems with integers.
Multiplying Integers
Long ago someone decided that instead of writing 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 32,
we could write 8 ∙ 4 = 32. Today we use this operation, called multiplication, so often that we
memorize multiplication facts.
Once we know that 8 ∙ 4 = 32, we also know that 4 ∙ 8 = 32. This is the commutative
property of multiplication. We get two math facts with just one memory. Multiplication is very
efficient.
When we add together a group of negative integers, we still multiply. So instead of
writing −3 + −3 + −3 + −3 + −3 = −15, we write 5 ∙ (−3) = −15. Using the
commutative property for multiplication we can write (−3) ∙ 5 = −15.1
Example 1: Write the multiplication equation that fits each addition. Then use the
commutative property to write another multiplication equation that also fits the addition.
Addition Equation
Multiplication Equation
a.
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 16
b.
(−7) + (−7) + (−7) + (−7) + (−7) = −35
5 ∙ (−7) = (−7) ∙ 5 = −35
c.
(−6) + (−6) + (−6) + (−6) = −24
4 ∙ (−6) = (−6) ∙ 4 = −24
1
8∙2 =
The properties listed in Section 1.4 on page 31 are true for all real numbers, including the integers.
2∙8 =
16
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SECTION 2.4 Multiplying Integers
 Check Point 1
Addition Equation
a.
−2 + −2 + −2 + −2 + −2 = −10
b.
(−3) + (−3) + (−3) + (−3) = −12
c.
(−6) + (−6) + (−6) + (−6) + (−6) = −30
Multiplication Equation
The pattern that you see in this example and your check point is always true; it is a
property of multiplying signed numbers.
PROPERTY: When two factors have different signs, their product is negative.
This property leads to a rule for multiplying. Mathematics is not a series of tricks
or a bunch of recipes that someone made up. All of the rules we use for
calculating are true because of what is always true about our numbers and operations. All rules
rely on our properties.
RULE: To multiply two numbers with different signs, multiply their absolute
values. The product is negative.
To determine what happens when we multiply (−6) by
a negative number, we make a table and look for a pattern.
We start by multiplying (−6) by positive numbers,
because we know that a negative times a positive is negative.
We decrease our second factor by one in each successive step
and let the pattern show us what to expect when the second
factor becomes negative.
What we find is that the product increases by six at
every step. Using this information, fill in the rest of the table.
UNIT TWO: Prealgebra in a Technical World
This pattern suggests that the product of two negative numbers is positive. If we use a
different negative factor, if we had replaced −6 with a different negative number, this pattern
would still be the same. If we make the table longer and include negative numbers that are
farther from zero, the product would still be positive. Try this! Draw your own tables. This
pattern gives us another rule for calculation.
RULE: To multiply two numbers with the same sign, multiply their absolute
values. The product is always positive.
Most of us have become skillful at avoiding negative numbers. We even avoid using
negative numbers in writing our budgets. But what happens in budgets can help make sense of
multiplying integers. Here is a story:
When Greg went to college, he kept a ledger of his income and his bills. His rent bill was
$350 per month. In his budget, all bills were recorded as negatives. His rent was recorded as
−$350 each month.
For Christmas, Greg’s grandmother sent four checks of $350 to pay for four months of
rent. Greg records this in his ledger. His rent is still negative, and his grandmother has taken
away his rent payment four times (−4). In his budget he records the four times his rent bill is
taken away as (−4) ∙ (−$350), and this results in an increase of $1,400.
Many people would have recorded Greg’s present as an increase, 4 ∙ $350. Since
(−4) ∙ (−350) and 4 ∙ 350 both give the same product, $1,400, both ledgers would balance.
By “taking away negatives,” Greg’s grandmother added $1,400 to Greg’s budget.
Some people come to understand that “a negative times a negative is a positive” by
using language. While two negatives make a positive in English, we avoid using these “double
negatives” because such statements are easily misunderstood. For instance, “I don't have no
money,” does not mean I am broke. It means that I do have some money. Multiplying two
negative numbers is like using double negatives: they multiply to a positive.
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SECTION 2.4 Multiplying Integers
Be careful not to just memorize rules without understanding what they mean.
A common error is to say that "two negatives make a positive." If you are
multiplying, this is true. But this is not true for addition! We now know that
(−3) ∙ (−4) = 12, but adding two negatives always results in a negative,
(−3) + (−4) = −7.
Example 2: Tell whether your product is positive (+) or negative (−) first, then multiply.
Sign of
Answer
−
Product
a.
Multiplication
Fact
(−𝟓) ∙ 𝟗
b.
(−𝟑) ∙ (−𝟕)
+
21
e.
c.
𝟕 ∙ (−𝟖)
−
−56
f.
−45
Multiplication
Fact
d. (−𝟐) ∙ (−𝟒)
Sign of
Answer
+
Product
(−𝟔) ∙ (−𝟕)
+
42
𝟑 ∙ (−𝟐)
−
−6
8
 Check Point 2
Tell whether your product is positive (+) or negative (−) first, then multiply.
a.
Multiplication
Fact
𝟕 ∙ (−𝟏)
Sign of
Answer
Product
Multiplication
Fact
(−𝟔)
∙ (−𝟐)
d.
b.
(−𝟓) ∙ 𝟖
e.
𝟑 ∙ (−𝟗)
c.
(−𝟒) ∙ (−𝟖)
f.
(−𝟖) ∙ (−𝟗)
Sign of
Answer
Product
Take a minute to sort out the three operations we have discussed so far. Fill in the
table. Check your work using the mixed answers on the right.
Adding, Subtracting, and Multiplying: Pay Attention to Signs
Mixed Answers
(−𝟒) + (−𝟖) =
(−𝟒) − (−𝟖) =
(−𝟒) ∙ (−𝟖) =
𝟕 ∙ (−𝟏) =
𝟕 + (−𝟏) =
𝟕 − (−𝟏) =
6, -7, 8
(−𝟓) − 𝟖 =
(−𝟓) ∙ 𝟖 =
(−𝟓) + 𝟖 =
-40, 3, -13
(−𝟖) ∙ (−𝟗) =
(−𝟖) − (−𝟗) =
(−𝟖) + (−𝟗) =
-17, 1, 72
𝟑 − (−𝟗) =
𝟑 + (−𝟗) =
𝟑 ∙ (−𝟗) =
-6, 12, -27
(−𝟔) + (−𝟐) =
(−𝟔) ∙ (−𝟐) =
(−𝟔) − (−𝟐) =
-8, -4, 12
32, 4, -12
UNIT TWO: Prealgebra in a Technical World
Multiplication Properties of 0 and -1
The numbers -1 and 0 are interesting.
PROPERTIES: Let 𝑎 be any integer, then the following are true.
Name
Property
Example
Multiplication
0∙𝑎 =𝑎∙0=0
0 ∙ −5 = −5 ∙ 0 = 0
Multiplication
−1 ∙ 𝑎 = 𝑎 ∙ (−1) = −𝑎
−1 ∙ 4 = 4 ∙ −1 = −4
Property of -1
(Remember: – 𝑎 is the opposite of 𝑎;
it is not always negative.)
Property of Zero
 Check Point 3
Use the properties of numbers to simplify these expressions.
a. (−1)(457) = ___________________ b. (−1)(−1)(−1) = ___________________
c. (−1)(0)(24) =__________________ d. (−100)(65)(−1) = _________________
e. (−32)(−1) = ___________________ f. (−1)(−1)(−1)(−1) = _______________
Simplifying Expressions
The last two problems in Check Point 3 may suggest what happens when we multiply
several signed numbers together.
Example 3: Simplify each of the following.
a. (−1)(−2)
b. (−2)(−2)(−3)(1)(−2)
c. (−1)(−5)(−2)(−3)(−2)
d. (−10)(−6)(−8)(−1)
e. (−3)(−3)(−5)
f. (5)(2)(−1)(−5)(−1)
ANSWERS:
a. 2
b. 24
c. −60
d. 480
e. −45
f. −50
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SECTION 2.4 Multiplying Integers
 Check Point 4
Complete the following sentences using the sample above.
a. If the number of negative factors in an expression is even, then the product is __________.
b. If the number of negative factors in an expression is odd, then the product is __________.
In Check Point 4, just above, you completed the rules for multiplying negative factors.
Highlight these; they are the rules!
Simplifying Expressions containing Exponents
Remember that we refer to exponential expressions as a base raised to a power:
BASE(POWER) or a base raised to an exponent: BASE(EXPONENT). The base is a factor. The power is
the exponent and tells how many times the base factor is multiplied. For example, in the
expression 85, 8 is the base and we call 5 either the “exponent” or “power”; 85 means 8 ∙ 8 ∙
8 ∙ 8 ∙ 8.
When we raise a positive base to any power, the answer is always positive. For example
25 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 32. Since all factors are positive, the product is positive.
When we raise a negative base to a power, the sign of the answer depends on whether
the exponent is even or odd. When the base is negative, we need to be careful. See the table:
RULE: The result of a negative base raised to a power is:

Positive if the power is even, because an even number of negative
numbers are multiplied together.

Negative if the power is odd, because an odd number of negative
numbers are multiplied together.
UNIT TWO: Prealgebra in a Technical World
In addition, we often find problems where we must find the opposite of a base raised to
a power. In these problems we must sort through several dashes! Here we carefully analyze a
few results.
A Base Raised to a Power
32
=
3∙3 =
+𝟗
2
(−3)(−3)
(−3) =
= +𝟗
53
=
5 ∙ 5 ∙ 5 = +𝟏𝟐𝟓
(−2)4 = [(−2)(−2)(−2)(−2)] = +𝟏𝟔
[(−1)(−1)(−1)] = −𝟏
(−1)3 =
4
(−1) = [(−1)(−1)(−1)(−1)] = +𝟏
The Opposite of the Same Base Raised
to the Same Power
2
−3 =
−(3 ∙ 3) =
−𝟗
2
−(−3) =
−[(−3)(−3)] = −𝟗
−53 =
−(5 ∙ 5 ∙ 5) = −𝟏𝟐𝟓
4
−(−2) = −[(−2)(−2)(−2)(−2)] = −𝟏𝟔
−(−1)3 =
−[(−1)(−1)(−1)] = +𝟏
−(−1)4 = −[(−1)(−1)(−1)(−1)] = −𝟏
Carefully notice the difference between a negative number squared, like (−3)2 , and
the opposite of a number squared, like −32 . A negative number squared has a positive result.
The opposite of any number squared gives a negative result. This is important math grammar
that you will want to make sense of and remember.
For all non-zero numbers 𝑎, (−𝑎)2 is always positive, and −𝑎2 is always
negative. This is true for all even powers, not just the exponent 2.
.
For instance, (−3)4 = 81 = 34 and – (−3)4 = −81 = −34 . Because 81 ≠
−81, (−3)4 𝑐𝑎𝑛 𝑛𝑒𝑣𝑒𝑟 𝑒𝑞𝑢𝑎𝑙 −34 . In the same way (−𝑎)2 ≠ −𝑎2 .
Using correct vocabulary can help sort out the meaning of these expressions.
Example 4: Say the expression out loud using correct vocabulary, then determine the sign,
then simplify.
a.
𝟕𝟐
b.
(−𝟕)𝟐
c.
−𝟕𝟐
d. −(−𝟕)𝟐
SAY: “7 squared”; the sign is (+).
SIMPLIFIED: 49
SAY: “negative 7 squared”; the sign is (+).
SIMPLIFIED: 49
SAY: “the opposite of 7 squared”; the sign is (−).
SIMPLIFIED: −49
SAY: “the opposite of negative 7 squared”; sign is (−).
SIMPLIFIED: −49
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SECTION 2.4 Multiplying Integers
 Check Point 5
Say these to yourself using correct vocabulary, determine the sign, then simplify.
a. (−5)3 = ______
b. −(−5)3 = ______
c. −53 = ______
d. (−4)2 = ______
e. −(−4)2 = ______
f. −42 = ______
g. (−3)4 = ______
h. −(−3)4 = ______
i. −34 = ______
j. 34
k. −34
l. −(3)4 = ______
= ______
= ______
Multiplication of Integers in Applications
Formulas are used when we solve the same type of problem repeatedly. We even have
formulas that give estimated results. For instance we can find the approximate Fahrenheit
temperature when given Celsius temperature by using the estimation formula 𝐹 ≈ 2𝐶 + 30.
The “≈” sign is read “is approximately equal to.”
People in the Far North often face temperatures below zero during the winter months.
They use this estimation formula to find the approximate Fahrenheit temperature when they
know the Celsius temperature.
Example 5: On their way north to Alaska, Ian and Addison plan to spend a night in Whitehorse,
Yukon Territory, Canada. When they wondered just how cold it might be, they checked the
Canadian government climate Web site2. Here they found that in January, the month that they
plan to drive through, the average temperature in Whitehorse is −22°C.
What is this temperature, −22°C, in degrees Fahrenheit?
Think it through: Since 𝑭 ≈ 𝟐𝑪 + 𝟑𝟎, the average Fahrenheit temperature is 𝟐(−𝟐𝟐) + 𝟑𝟎.
ANSWER: The average temperature for January in Whitehorse is about −𝟏𝟒°F.
2
Environment Canada, National climate data information archive.
http://www.climate.weatheroffice.ec.gc.ca/climate_normals/index_e.html, accessed July 31, 2009.
UNIT TWO: Prealgebra in a Technical World
 Check Point 6
The lowest temperature ever recorded in Whitehorse was −52°C, and it occurred during
January 1947. Use the estimated formula to find the approximate Fahrenheit temperature for
−52°C. _______________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Sentence:______________________________________________________________________
______________________________________________________________________________
 Check Point 7
Steve is trying to put together an annual budget. His expenses are recorded as negative
numbers, while his income is recorded as positive. What is the total he records for 3 tuition
bills that are $1,350 each? ________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Sentence:______________________________________________________________________
______________________________________________________________________________
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SECTION 2.4 Multiplying Integers
UNIT TWO: Prealgebra in a Technical World
2.4 Exercise Set
Name _______________________________
Skills
Determine the sign of each product first, then multiply.
1. −5 ∙ 2
2. 4 ∙ (−4)
3. −5 ∙ 4
4. 2 ∙ (−4)
5. 7 ∙ (−7)
6. −4 ∙ 4
7. −10 ∙ (−4)
8. −5 ∙ (−7)
9. −21 ∙ (−50)
13. 89 ∙ (−45)
10. −36 ∙ (−5)
14.
−27 ∙ (−35)
11. 4 ∙ (−23)
12. −6 ∙ 37
15. 6 ∙ (−2)
16. −24 ∙ 24
17.
124 ∙ (−35)
18. 14 ∙ (−15)
19. −11 ∙ (−11)
20. −20 ∙ (−20)
21.
492 ∙ (−13)
22. 5 ∙ (−11)
23. −9 ∙ 24
24. −17 ∙ 19
25. −7 ∙ 3
26. −1 ∙ (−6)
27. −111 ∙ (−40)
28. −8 ∙ 3
29. 13 ∙ (−12)
30. 15 ∙ (−7)
31. −29 ∙ (−41)
32. 14 ∙ (−63)
33. (−3) ∙ (−4) ∙ (−5)
34. (−1) ∙ (−1) ∙ (−1)
35. 5 ∙ (−3)(−4)
36. 11 ∙ (−11) ∙ (−2)
37. 7 ∙ (−2) ∙ 5
38. −6 ∙ (−10) ∙ (4)
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SECTION 2.4 Multiplying Integers
39. −1(4)(−2)
40. 12(−30)(−5)
41. −2(−16)(−5)
42. 17(14)(−3)
43. −14(−17)(2)
44. 4(−4)(4)
45. −13(7) =
46. −237(436)(0)(49)
47.
−15(−6)(0)
Remember that −32 is read, "the opposite of 3 squared," and −32 = −9. Remember too that
(−3)2 is read, "negative 3 squared," and (−3)2 = 9. The base is 3 in the first case, while the
base is −3 in the second case. Simplify the following.
−52 =________
49. (−4)2 =________
50.
−33 =________
51. −(9)2 =________
52. (−2)2 =________
53.
−62 =________
54. (−1)4 =________
55.
57. −(5)3 =________
58. (−5)3 =________
60. (−1)6 =________
61.
48.
−72 =________
−71 =________
56. (−8)2 =________
59.
−15 =________
62. (−8)1 =________
Remember that a variable can be either positive or negative. Choose negative or positive to
make the following sentences true.
63. If 𝑥 < 0, 𝑥 2 is ALWAYS
negative / positive. (Circle One)
64. If 𝑥 > 0, −𝑥 2 is ALWAYS negative / positive. (Circle One)
65. If 𝑥 < 0, 𝑥 3 is ALWAYS
negative / positive. (Circle One)
66. If 𝑥 > 0, −𝑥 4 is ALWAYS negative / positive. (Circle One)
67. If 𝑥 < 0, 𝑥 4 is ALWAYS
negative / positive. (Circle One)
68. If 𝑥 > 0, −𝑥 3 is ALWAYS negative / positive. (Circle One)
UNIT TWO: Prealgebra in a Technical World
Applications UPS
69.
71.
The temperature gauge in a small town
in Alaska measured 0℉ one Friday
evening. The temperature dropped an
average of 5°every night after that day
for a week. What was the temperature
on the following Friday?
70. One winter Monday night in a village in
Siberia, Olga recorded a low
temperature of −4℉. By Friday, the
temperature was 10 times colder. How
cold was it on Friday?
Sentence:_________________________
Sentence:_________________________
_________________________________
_________________________________
_________________________________
_________________________________
Chrissy runs an Internet business selling silk-screened t-shirts for bird-lovers. She sells the
t-shirts for $22 each, including shipping and handling. She guarantees that you will be
satisfied with her product, or you will receive a full refund. After Christmas, she had 13
of her t-shirts returned for a refund. She recorded this debit/credit as
________________in her ledger.
Review and Extend
Practice the paper and pencil multiplication algorithm. Write the sign of your answer first.
72.
−524
x 31
73.
−634
x −75
74.
−1004
x 708
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SECTION 2.4 Multiplying Integers
Translate the following English expressions and sentences into numeric expressions, equations
and inequalities. Then simplify each.
75.
The product of negative seven, negative three and five
a. translated: ___________________________ b. simplified: ____________________
76.
The product of negative seven and negative eight, plus five
a. translated: ___________________________ b. simplified: ____________________
77.
The product of negative seven and the quantity negative eight plus five
a. translated: ___________________________ b. simplified: ____________________
78.
The sum of negative seven and negative three times five
a. translated: ___________________________ b. simplified: ____________________
79.
The difference of negative eleven and negative four, times twenty
a. translated: ___________________________ b. simplified: ____________________
80.
Negative eleven times the difference of four and twenty
a. translated: ___________________________ b. simplified: ____________________
81.
Negative eight minus seven is less than negative eight plus seven.
a. translated: ___________________________ b. simplified: ____________________
82.
The difference of two and, the product of negative six and seven is forty-four.
a. translated: ___________________________ b. simplified: ____________________
83.
Negative eleven minus seven is the same as the opposite of the product of three and six.
a. translated: ___________________________ b. simplified: ____________________
84.
The absolute value of the quantity seven minus forty is less than negative three cubed.
a. translated: ___________________________ b. simplified: ____________________