Basic Mathematics for College Students

The properties of multiplication that were introduced in Section 1.3, Multiplying

Whole Numbers, are also true for integers.

168 Chapter 2 The Integers

EXAMPLE (^3) Evaluate each expression:
a. b. c.
StrategySince there are no calculations within parentheses and no exponential
expressions, we will perform the multiplications, working from the left to the right.
WHYThis is step 3 of the order of operations rule that was introduced in Section 1.9.
Solution
a. Use the rule for multiplying two integers that have
different signs: 6(2) 12.
Use the rule for multiplying two integers that have
the same sign.
b. Use the rule for multiplying two integers that have different
signs: 9(8) 72.
Use the rule for multiplying two integers that have the same
sign.
c. Use the rule for multiplying two integers that have
the same sign: 3(5) 15.
Use the rule for multiplying two integers that have
the same sign: 15(2) 30.
Use the rule for multiplying two integers that have
different signs.

 120

30(4)

3(5)(2)(4) 15 (2)(4)

 72

9(8)(1) 72 (1)

 84

6(2)(7) 12 (7)

6(2)(7) 9(8)(1) 3(5)(2)(4)

Self Check 3

Evaluate each expression:

a.

b.

c.

Now TryProblems 45, 47, and 49

4(5)(8)(3)

1(9)(6)

3(12)(2)

Self Check 4

Use the commutative and/or

associative properties of

multiplication to evaluate each

expression from Self Check 3

in a different way:

a.

b.

c.

Now TryProblems 45, 47, and 49

4(5)(8)(3)

1(9)(6)

3(12)(2)

Properties of Multiplication

Commutative property of multiplication: The order in which integers are

multiplied does not change their product.

Associative property of multiplication: The way in which integers are

grouped does not change their product.

Multiplication property of 0: The product of any integer and 0 is 0.

Multiplication property of 1: The product of any integer and 1 is that integer.

Another approach to evaluate expressions like those in Example 3 is to use the

properties of multiplication to reorder and regroup the factors in a helpful way.

EXAMPLE (^4) Use the commutative and/or associative properties of
multiplication to evaluate each expression from Example 3 in a different way:
a. b. c.
StrategyWhen possible, we will use the commutative and/or associative
properties of multiplication to multiply pairs of negative factors.
WHYThe product of two negative factors is positive. With this approach, we work
with fewer negative numbers, and that lessens the possibility of an error.
Solution
a. Multiply the last two negative factors to produce a
positive product: 7(2) 14.
 84

6 (2)(7) 6 (14)

6(2)(7) 9(8)(1) 3(5)(2)(4)

1

2

4

 6

84

1

1

2

 7

84