38
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
(1-38)
Chapter 1
The Real Numbers
3(xy) (3x)y Associative property of multiplication
3(x 1) 3x 3 Distributive property
4(0.25) 1 Multiplicative inverse property
0.3 9 9 0.3 Commutative property of addition
y3x xy3 Commutative property of multiplication
0 52 0 Multiplication property of zero
1 x x Multiplicative identity property
(0.1)(10) 1 Multiplicative inverse property
2x 3x (2 3)x Distributive property
8 0 8 Additive identity property
7 (7) 0 Additive inverse property
1 y y Multiplicative identity property
(36 79)0 0 Multiplication property of zero
5x 5 5(x 1) Distributive property
xy x x(y 1) Distributive property
ab 3ac a(b 3c) Distributive property
Complete each statement using the property named.
81. 5 w _____, commutative property of addition
w5
82. 2x 2 _____, distributive property
2(x 1)
83. 5(xy) ____, associative property of multiplication
(5x)y
1
84. x _____, commutative property of addition
2
1
x
2
1.6
In this
section
1
1
1
85. x _____, distributive property (x 1)
2
2
2
86. 3(x 7) _____, distributive property 3x 21
87. 6x 9 _____, distributive property 3(2x 3)
88. (x 7) 3 _____, associative property of addition
x (7 3)
89. 8(0.125) _____, multiplicative inverse property 1
90. 1(a 3) _____, distributive property a 3
91. 0 5(_____), multiplication property of zero 0
92. 8 (_____) 8, multiplicative identity property 1
93. 0.25 (_____) 1, multiplicative inverse property 4
94. 45(1) _____, multiplicative identity property 45
GET TING MORE INVOLVED
95. Discussion. Does the order in which your groceries are
placed on the checkout counter make any difference in your
total bill? Which properties are at work here?
96. Discussion. Suppose that you just bought 10 grocery items
and paid a total bill that included 6% sales tax. Would there
be any difference in your total bill if you purchased the items
one at a time? Which property is at work here?
USING THE PROPERTIES
The properties of the real numbers can be helpful when we are doing computations.
In this section we will see how the properties can be applied in arithmetic and
algebra.
●
Using the Properties in
Computation
●
Like Terms
●
Combining Like Terms
Consider the product of 36 and 200. Using the associative property of multiplication, we can write
●
Products and Quotients
(36)(200) (36)(2 100) (36 2)(100).
●
Removing Parentheses
E X A M P L E
Using the Properties in Computation
To find this product mentally, first multiply 36 by 2 to get 72, then multiply 72 by
100 to get 7200.
1
Using properties in computation
Evaluate each expression mentally by using an appropriate property.
1
c) 7 45 3 45
a) 536 25 75
b) 5 426 5
1.6
study
tip
Find out what help is available
at your school. Accompanying this text are video tapes,
solution manuals, and a computer tutorial. Around most
campuses you will find tutors
available for hire, but most
schools have a math lab
where you can get help for
free. Some schools even
have free one-on-one tutoring available through special
programs.
Using the Properties
(1-39) 39
Solution
a) To perform this addition mentally, the associative property of addition can be
applied as follows:
536 (25 75) 536 100 636
b) Use the commutative and associative properties of multiplication to rearrange
mentally this product.
1
1
5 426 426 5 5
5
1
426 5 5
426 1
426
Commutative property of multiplication
Associative property of multiplication
Multiplicative inverse property
c) Use the distributive property to rewrite the expression, then evaluate it.
7 45 3 45 (7 3)45 10 45 450
■
Like Terms
The properties of the real numbers are used also with algebraic expressions. Simple
algebraic expressions such as
2,
4x,
5x2y,
b,
and
abc
are called terms. A term is a single number or the product of a number and one or
more variables raised to powers. The number preceding the variables in a term is
called the coefficient. In the term 4x the coefficient of x is 4. In the term 5x 2y the
coefficient of x 2y is 5. In the term b the coefficient of b is 1, and in the term abc
the coefficient of abc is 1. If two terms contain the same variables with the same
powers, they are called like terms. For example, 3x 2 and 5x 2 are like terms,
whereas 3x 2 and 2x 3 are not like terms.
Combining Like Terms
We can combine any two like terms involved in a sum by using the distributive
property. For example,
2x 5x (2 5)x
7x
Distributive property
Add 2 and 5.
Because the distributive property is valid for any real numbers, we have 2x 5x 7x for any real number x.
We can also use the distributive property to combine any two like terms involved in a difference. For example,
3xy (2xy) [3 (2)]xy
1xy
xy
Distributive property
Subtract.
Multiplying by 1 is the same as taking the opposite.
Of course, we do not want to write out these steps every time we combine like terms.
We can combine like terms as easily as we can add or subtract their coefficients.
40
(1-40)
Chapter 1
E X A M P L E
study
2
tip
Read the material in the text
before it is discussed in class,
even if you do not totally understand it. The classroom discussion will be the second
time you see the material and
it will be easier then to question points that you do not
understand.
The Real Numbers
Combining like terms
Perform the indicated operation.
a) b 3b
c) 5xy (13xy)
b) 5x 2 7x 2
d) 2a (9a)
Solution
a) b 3b 1b 3b 4b
c) 5xy (13xy) 18xy
b) 5x 2 7x 2 2x 2
d) 2a (9a) 11a
■
The distributive property allows us to combine only like terms.
CAUTION
Expressions such as
3xw 5,
7xy 9t,
5b 6a,
and
6x 2 7x
do not contain like terms, so their terms cannot be combined.
Products and Quotients
We can use the associative property of multiplication to simplify the product of two
terms. For example,
4(7x) (4 7)x
(28)x
28x
Associative property of multiplication
Remove unnecessary parentheses.
CAUTION
Multiplication does not distribute over multiplication. For
example, 2(3 4) 6 8 because 2(3 4) 2(12) 24.
helpful
hint
Did you know that the line
separating the numerator and
denominator in a fraction is
called the vinculum?
In the next example we use the fact that dividing by 3 is equivalent to multiplying by 1, the reciprocal of 3:
3
x
1
3 3 x
3
3
1
3 x
3
Definition of division
Commutative property of multiplication
1
3 x
3
Associative property of multiplication
1x
1
3 1 (Multiplicative inverse property)
3
x
Multiplicative identity property
To find the product (3x)(5x), we use both the commutative and associative properties of multiplication:
(3x)(5x) (3x 5)x
(3 5x)x
(3 5)(x x)
(15)(x 2)
15x 2
Associative property of multiplication
Commutative property of multiplication
Associative property of multiplication
Simplify.
Remove unnecessary parentheses.
1.6
Using the Properties
(1-41) 41
All of the steps in finding the product (3x)(5x) are shown here to illustrate that
every step is justified by a property. However, you should write (3x)(5x) 15x 2
without doing any intermediate steps.
E X A M P L E
3
Multiplying terms
Find each product.
a) (5)(6x)
b) (3a)(8a)
c) (4y)(6)
b
d) (5a) 5
Solution
a) 30x
c) 24y
d) ab
b) 24a2
■
In the next example we use the properties to find quotients. Try to identify the
property that is used at each step.
E X A M P L E
study
4
tip
Ask questions in class. If you
don’t ask questions, then the
instructor might believe that
you have total understanding.
When one student has a
question, there are usually
several who have the same
question but do not speak up.
Asking questions not only
helps you to learn, but it keeps
the classroom more lively and
interesting.
Dividing terms
Find each quotient.
5x
a) 5
4x 8
b) 2
Solution
a) First use the definition of division to change the division by 5 to multiplication
by 1.
5
5x
1
1
5x 5 x 1 x x
5
5
5
b) First use the definition of division to change division by 2 to multiplication
by 12.
4x 8
1 1
(4x 8) (4x 8) 2x 4
2
2 2
Since both 4x and 8 are divided by 2, we could have written
4x 8 4x 8
2x 4.
2
2
2
CAUTION
■
Do not divide a number into just one term of a sum. For
example,
27
17
2
because
27 9
2
2
and
1 7 8.
42
(1-42)
Chapter 1
The Real Numbers
Removing Parentheses
Multiplying a number by 1 merely changes the sign of the number. For example,
(1)(6) 6
and
(1)(15) 15.
Thus 1 times a number is the same as the opposite of the number. Using variables,
we have
(1)x x
or
1(a 2) (a 2).
When a minus sign appears in front of a sum, we can think of it as multiplication
by 1 and use the distributive property. For example,
(a 2) 1(a 2)
(1)a (1)2
a (2)
a 2.
Distributive property
If a minus sign occurs in front of a difference, we can rewrite the expression as a
sum. For example,
(x 5) 1(x 5) (1)x (1)5 x 5.
Note that a minus sign in front of a set of parentheses affects each term in the parentheses, changing the sign of each term.
E X A M P L E
5
Removing parentheses
Simplify each expression.
a) 6 (x 8)
b) 4x 6 (7x 4)
Solution
a) 6 (x 8) 6 x 8
68x
2 x
c) 3x (x 7)
Change the sign of each term in parentheses.
Rearrange the terms.
Combine like terms.
b) 4x 6 (7x 4) 4x 6 7x 4 Remove parentheses.
4x 7x 6 4 Rearrange the terms.
3x 2
Combine like terms.
c) 3x (x 7) 3x x 7 Remove parentheses.
4x 7
Combine like terms.
■
The commutative and associative properties of addition allow us to rearrange
the terms so that we may combine like terms. However, it is not necessary actually
to write down the rearrangement. We can identify like terms and combine them
without rearranging.
E X A M P L E
6
More parentheses and like terms
Simplify each expression.
a) (5x 7) (2x 9)
c) 3x(4x 9) (x 5)
b) 4x 7x 3(2 5x)
d) x 0.03(x 300)
1.6
Using the Properties
(1-43) 43
Solution
a) (5x 7) (2x 9) 3x 2 Combine like terms.
b) 4x 7x 3(2 5x) 4x 7x 6 15x Distributive property
12x 6
Combine like terms.
2
c) 3x(4x 9) (x 5) 12x 27x x 5 Remove parentheses.
12x 2 26x 5
Combine like terms.
d) x 0.03(x 300) 1x 0.03x 9 Distributive property; (0.03)(300) 9
0.97x 9
Combine like terms: 1.00 0.03 0.97
WARM-UPS
True or false? Explain your answer.
A statement involving variables should be marked true only if it is true for all
values of the variable.
1. 5(x 7) 5x 35 True
2. 4x 8 4(x 8) False
3. 1(a 3) (a 3) True
4. 5y 4y 9y True
5. (2x)(5x) 10x False
6. 2t(5t 3) 10t2 6t True
2
8. b b 2b False
7. a a a False
9. 1 7x 8x False
10. (3x 4) (8x 1) 5x 3 True
1. 6
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is a term?
A term is a single number or a product of a number and one
or more variables.
2. What are like terms?
Like terms contain the same variables with the same powers.
3. What is the coefficient of a term?
The coefficient of a term is the number preceding the
variables.
4. Which property is used to combine like terms?
The distributive property is used to combine like terms.
5. What operations can you perform with unlike terms?
You can multiply and divide unlike terms.
6. How do you remove parentheses that are preceded by a
negative sign?
You can remove parentheses preceded by a negative sign
by taking the opposite of every term in the parentheses.
Perform each computation. Make use of appropriate rules to
simplify each problem. See Example 1.
7. 45(200) 9,000
4
9. (0.75) 1
3
8. 25(300)
10. 5(0.2)
7,500
1
11. (427 68) 32 527
13. 47 4 47 6 470
1
15. 19 5 2 38
5
17. (120)(400) 48,000
19. 13 377(5 5)
21. (348 5) 45
2
23. (1.5) 1
3
25.
26.
27.
28.
29.
30.
0
398
12. (194 78) 22 294
14. 53 3 53 7 530
1
16. 17 4 2 34
4
18. 150 300 45,000
1
20. (456 8) 456
8
22. (135 38) 12 185
24. (1.25)(0.8)
1
17 101 17 1 1,700
33 2 12 33 330
354 7 8 3 2 374
564 35 65 72 28 764
(567 874)(2 4 8) 0
(5672 48)[3(5) 15] 0
Combine like terms where possible. See Example 2.
31. 4n 6n
32. 3a 15a
2n
12a
33. 3w (4w)
34. 3b (7b)
7w
10b
■
44
(1-44)
35. 4mw2 15mw2
11mw 2
37. 5x (2x)
3x
39. 4 7z
4 7z
41. 4t 2 5t 2
9t 2
43. 4ab 3a 2b
4ab 3a 2b
45. 9mn mn
8mn
47. x 3y 3x 3y
2x 3y
49. kz 6 kz 6
2kz6
Chapter 1
The Real Numbers
36. 2b2x 16b2x
14b 2x
38. 11 7t
11 7t
40. 19m (3m)
16m
42. 5a 4a 2
5a 4a 2
44. 7x 2y 5x 2y
2x 2y
46. 3cm cm
2cm
48. s 4t 5s 4t
4s 4t
50. m7w m7w
0
Find each product or quotient. See Examples 3 and 4.
51. 4(7t)
52. 3(4r)
53. (2x)(5x)
28t
12r
10x 2
54. (3h)(7h)
55. (h)(h)
56. x(x)
21h2
h2
x 2
57. 7w(4)
58. 5t(1)
59. x(1 x)
28w
5t
x x 2
60. p( p 1)
61. (5k)(5k)
62. (4y)(4y)
p2 p
25k 2
16y2
y
63. 3 3
y
y
66. 8 8
y
3x 2 y 15x
69. 3
x 2y 5x
6x 9
72. 3
2x 3
z
64. 5z 5
z2
6x 3
67. 2
3x 3
6xy 2 8w
70. 2
3xy 2 4w
xt 10
73. 2
1
xt 5
2
Simplify each expression. See Example 5.
75. a (4a 1) 3a 1
76. 5x (2x 7) 3x 7
77. 6 (x 4) 10 x
78. 9 (w 5) 14 w
79. 4m 6 (m 5) 3m 1
80. 5 6t (3t 4) 1 9t
81. 5b (at 7b) 12b at
82. 4x 2 (7x 2 2y) 3x 2 2y
83. t 2 5w (2w t 2) 2t 2 3w
84. n2 6m (n2 2m ) 2n2 4m
2y
65. 9 9
2y
8x2
68. 4
2x 2
2x 4
71. 2
x 2
2xt 2 8
74. 4
1 2
xt 2
2
85. x 2 (x 2 y 2 z) y 2 z
86. 5w (6w 3xy zy) 3xy zy w
Simplify each expression. See Example 6.
87. (2x 7x) (3 5) 9x 8
88. (3x 4x) (5 12) 7x 17
89. (3x 4) (5x 6) 2x 2
90. (4x 11) (6x 8) 2x 3
2
91. 4a2 5c (6a 7c) 2a2 2c
2
2
92. 3x 4 (x 5) 2x2 1
93. 5(t 2 3w) 2(3w t 2) 7t 2 9w
94. 6(xy2 2) 5(xy2 1) 11xy2 17
95. 7m 3(m 4) 5m m 12
96. 6m 4(m 3) 7m 5m 12
97. 8 7(k 3 3) 4 7k 3 17
98. 6 5(k 3 2) k 3 5 4k 3 1
99. x 0.04(x 50) 0.96x 2
100. x 0.03(x 500) 0.97x 15
101. 0.1(x 5) 0.04(x 50) 0.06x 1.5
102. 0.06x 0.14(x 200) 0.2x 28
103. 3k 5 2(3k 4) k 3 4k 16
104. 5w 2 4(w 3) 6(w 1) 3w 8
105. 5.7 4.5(x 3.9) 5.42 4.5x 17.83
106. 0.04(5.6x 4.9) 0.07(7.3x 34) 0.735x 2.576
Simplify.
107. 3(1 xy) 2(xy 5) (35 xy) 4xy 22
108. 2(x2 3) (6x2 2) 2(7x2 4) 18x2 12
109. w 3w 5w (6w) w(2w) 29w2
110. 3w3 5w3 4w 3 12w 3 2w 2 16w3 2w2
111. 3a2w2 5w2 a2 2aw 2aw 6a2w 2
112. 3(aw2 5a2w) 2(a2w a2w) 11a 2w 3aw2
1
1 1
1
113. 6x 2y 2x 2y 6 3
2
3
1
1
1
114. bc bc(3 a) abc 2bc
2
2
2
1
1
1
1
1
115. m m m m m2 m
2
2
2
2
4
4wyt
8wyt
2wy
116. 3wyt wy
4
2
2
8t 3 6t 2 2
117. 2
4t 3 3t 2 1
7x3 5x3 4
118. x3 2
2
6xyz 3xy 9z
119. 2xyz xy 3z
3
20a2b4 10a2b4 5
120. 5
2a2b4 1
Chapter 1
Write an algebraic expression for each problem.
121. Triangle. The lengths of the sides of a triangular flower
bed are s feet, s 2 feet, and s 4 feet. What is its
perimeter? 3s 6 ft
122. Parallelogram. The lengths of the sides of a lot in the
shape of a parallelogram are w feet and w 50 feet.
What is its perimeter? Is it possible to find the area from
this information? 4w 100 ft, no
Collaborative Activities
(1-45) 45
Parthenon is x meters and its length is x 1 x meters,
6
then what is the perimeter? What is the area?
13
7 2 2
x m, x m
3
6
124. Square. If the length of each side of a square sign
is x inches, then what are the perimeter and area of the
square? 4x in., x2 in.2
w + 50 ft
w ft
w ft
w + 50 ft
FIGURE FOR EXERCISE 122
123. Parthenon. To obtain a pleasing rectangular shape, the
ancient Greeks constructed buildings with a length that
was about 1 longer than the width. If the width of the
6
FIGURE FOR EXERCISE 124
COLLABORATIVE ACTIVITIES
OOOP! Order of Operations Game
Grouping: Three to five students per group
Topic: Order of operations, learning to work in groups
In this game you will be reviewing the established order of operations for real numbers. You may evaluate any compound expression while playing this game. Use a new piece of paper for
each game.
3. Each Player’s Task
1. Before Play Begins
You will need three to five players on a team. Assign each player
a role or operation: E—Exponents, M—Multiply/Divide (may
have two players working as a team, one who multiplies and one
who divides); A—Add/Subtract (may have two players working as a team, one who adds and one who subtracts).
2. Determining a Player’s Turn at Play
All players working together analyze the expression and
decide which part to complete. If there are parentheses, then
the players decide whether what is inside the parentheses
needs simplification. E performs his or her operation before
M, and M performs his or her operation before A. Each
player’s turn ends when he or she encounters an operation that
precedes his or hers or when he or she reaches the end of the
expression.
E—Exponents: Working left to right, E evaluates exponential expressions in order.
M—Multiply/Divide: Working left to right, M performs
multiplications and divisions in order. Multiplication and
division may be done by two players working as a team. If
these tasks are split, then the two players perform their assigned operations in order, taking turns as needed.
A—Add/Subtract: Working left to right, A performs additions and subtractions in order. Addition and subtraction
may be done by two players working as a team. If these
tasks are split, then the two players perform their assigned
operations in order, taking turns as needed.
4. Recording Results
Results are recorded on one sheet of paper. As each player finishes his or her operation, she or he passes the paper to the
player with the next task. Each player rewrites the new form of
the expression on the next line of the page and initials his or her
work with E, M, or A.