SECT
ION
0.5
0.5
0.5
OBJECTIVES
1. Plot rational numbers on a number
line
2. Evaluate numbers
with exponents
3. Use the order of
operations
4. Write a product
of like factors in
exponential form
Example 1
Exponents and Order
of Operations
“The intelligent person is one who has successfully fulfilled many accomplishments,
and yet is willing to learn more.”
–Ed Parker
To this point, every number that we have encountered in this text could have been written as a ratio of two integers.
3
1
25
For example, 3 could be written as or 3 could have been written as .
1
8
8
Such numbers are called rational numbers. On the number line, you can estimate the
location of a rational number, as our next example illustrates.
Plotting Rational Numbers
Plot each of the following rational numbers on the number line provided.
2
1 27
, 3, , 1.445
3
4 5
3 14
Note that a decimal is really
a “decimal fraction.” 1.445
is another way of writing
1.445
27
5
2
3
0
✓ CHECK YOURSELF 1
■
Plot each of the following rational numbers on the number line provided.
1445
1000
1 37
1
2, , 5.66, 3 11
4
4
2
0
2
4
6
In Section 0.4, we mentioned that multiplication is a more compact, or “shorthand,”
form for repeated addition. For example, an expression with repeated addition, such as
42
Section 0.5
■
Exponents and Order of Operations
43
33333
can be rewritten as
53
Thus multiplication is shorthand for repeated addition.
In algebra, we frequently have a number or variable that is repeated in an expression several times. For instance, we might have
A factor is a number or a
variable that is being
multiplied by another
number or variable.
Since an exponent represents
repeated multiplication, 53 is
an expression.
555
To abbreviate this product, we write
5 5 5 53
This is called exponential notation or exponential form. The exponent or power, here
3, indicates the number of times that the factor or base, here 5, appears in a product.
!
Caution
Exponent or power
5 5 5 53
3
Be careful: 5 is not the same
as 5 3. Notice that
53 5 5 5 125 and
5 3 15.
Example 2
Factor or base
Writing Expressions in Exponential Form
(a) Write 3 3 3 3, using exponential form. The number 3 appears 4 times in the
product, so
Four factors of 3
3 3 3 3 34
This is read “3 to the fourth power.”
(b) Write 10 10 10 using exponential form. Since 10 appears three times in the
product, you can write
10 10 10 103
This is read “10 to the third power” or “10 cubed.”
✓ CHECK YOURSELF 2
■
Write in exponential form.
(a) 4 4 4 4 4 4
(b) 10 10 10 10
44
Chapter 0
■
The Arithmetic of Signed Numbers
When evaluating a number raised to a power, it is important to note whether there
is a sign attached to the number. Note that
(2)4 (2)(2)(2)(2) 16
whereas,
24 (2)(2)(2)(2) 16.
Example 3
Evaluating Exponential Terms
Evaluate each expression.
(a) (3)3 (3)(3)(3) 27
(b) 33 (3)(3)(3) 27
(c) (3)4 (3)(3)(3)(3) 81
(d) 34 (3)(3)(3)(3) 81
✓ CHECK YOURSELF 3
■
Evaluate each expression.
(a) (4)3
(b) 43
(c) (4)4
(d) 44
Your calculator can help you to evaluate expressions containing exponents. If you
have a graphing calculator, the appropriate key is the carat, “.” Enter the base, followed by the carat, followed by the exponent. Other calculators use a key labeled “yx”
in place of the carat.
Example 4
Evaluating Expressions with Exponents
Use your calculator to evaluate each expression.
(a) 35 243
or 3
(b) 210 1024
Type 3
2
yx
or 2
5
5
Enter
10 Enter
yx
10
Section 0.5
■
Exponents and Order of Operations
45
✓ CHECK YOURSELF 4
■
Use your calculator to evaluate each expression.
(a) 34
(b) 216
We have used the term “expression” with numbers taken to powers, like 34. But
what about something like 4 12 6? We call any meaningful combination of numbers and operations an expression. When we evaluate an expression, we find a number that is equal to the expression. To evaluate an expression, we need to establish a
set of rules that tell us the correct order in which to perform the operations. To see
why, simplify the expression 5 2 3.
Only one of these results can
be correct.
Method 2
523
523
Add first.
Multiply first.
73
56
21
11
{
!
or
{
Caution
Method 1
Since we get different answers depending on how we do the problem, the language of
algebra would not be clear if there were no agreement on which method is correct. The
following rules tell us the order in which operations should be done.
Parentheses and brackets are
both grouping symbols.
Fraction bars and radicals
are also grouping symbols.
The Order of Operations
Step 1
Evaluate all expressions inside grouping symbols first.
Step 2
Evaluate all expressions involving exponents.
Step 3
Do any multiplication or division in order, working from left to right.
Step 4
Do any addition or subtraction in order, working from left to right.
46
Chapter 0
■
The Arithmetic of Signed Numbers
Example 5
Evaluating Expressions
Evaluate 5 2 3.
There are no parentheses or exponents, so start with step 3: First multiply and then
add.
523
Multiply first.
Note: Method 2 shown on
the previous page is the
correct one.
56
Then add.
11
✓ CHECK YOURSELF 5
■
Evaluate the following expressions.
(a) 20 3 4
Example 6
(b) 9 6 3
Evaluating Expressions
Evaluate 5 32.
5 32 5 9
Evaluate the exponent first.
45
✓ CHECK YOURSELF 6
■
Evaluate 4 24.
Both scientific and graphing calculators correctly interpret the order of operations.
This is demonstrated in Example 7.
Example 7
Using a Calculator to Evaluate Expressions
Use your scientific or graphing calculator to evaluate each expression.
(a) 24.3 6.2 3.5
When evaluating expressions by hand, you must consider the order of operations. In
this case, the multiplication must be done first, then the addition. With a modern cal-
Section 0.5
■
Exponents and Order of Operations
47
culator, you need only enter the expression correctly. The calculator is programmed to
follow the order of operations.
Entering 24.3
6.2
3.5 Enter
yields the evaluation 46.
(b) (2.45)3 49 8000 12.2 1.3
As we mentioned earlier, some calculators use the carat () to designate exponents.
Others use the symbol xy (or yx).
Entering (
2.45 )
3
49
8000
12.2
1.3
or
2.45 )
yx
3
49
8000
12.2
1.3
(
yields the evaluation 30.56.
✓ CHECK YOURSELF 7
■
Use your scientific or graphing calculator to evaluate each expression.
(a) 67.89 4.7 12.7
(b) 4.3 55.5 (3.75)3 8007 1600
Operations inside grouping symbols are done first.
Example 8
Evaluating Expressions
Evaluate (5 2) 3.
Do the operation inside the parentheses as the first step.
(5 2) 3 7 3 21
Add.
✓ CHECK YOURSELF 8
■
Evaluate 4(9 3).
The principle is the same when more than two “levels” of operations are involved.
48
Chapter 0
■
Example 9
The Arithmetic of Signed Numbers
Evaluating Expressions



(a) Evaluate 4(2 7)3.
Add inside the parentheses first.
4(2 7)3 4(5)3
Evaluate the exponent.
4(2 7)3 4 125
Multiply.
4(2 7)3 500
(b) Evaluate 5(7 3)2 10.
Evaluate the expression inside the parentheses.
5(7 3)2 10 5(4)2 10
Evaluate the exponent.
5 16 10
Multiply.
80 10 70
Subtract.
✓ CHECK YOURSELF 9
■
Evaluate
(a) 4 33 8 (11).
(b) 12 4(2 3)2.
✓ CHECK YOURSELF ANSWERS
■
1.
2 13
14 0
37
11
2. (a) 46; (b) 104.
5.66
3.
(a) 64; (b) 64; (c) 256; (d) 256.
4.
(a) 81; (b) 65,536.
5.
(a) 8; (b) 11.
6. 64.
7.
(a) 8.2; (b) 190.92.
8. 24.
9. (a) 20; (b) 112.
E xercises
5. 35
6. 27
7. 75
8. 105
6
6
9. 8
10. 5
11. 9
12. 8
13. 16
14. 32
15. 512
16. 243
17. 1
18. 256
19. 25
20. 216
21. 6561
22. 7
23. 1000
24. 100
25. 1,000,000
26. 10,000,000
27. 128
■
0.5
In Exercises 1 to 4, plot each of the rational numbers on the number line.
3
1 32
1. , 1, , 4.335
4
4 13
4.335
1 14 0
3
4
2
1 15
2. , 2, , 4.156
3
3 7
2 13
32
13
7
5 35
3. , 1, , 5.156
8
6 14
5.156
1 56
0
7
8
0
2
3
15
7
4.156
3 1
17
4. , 1, , 3.165
5 8
8
17
8
35
14
0
3
5
118
3.165
Write each expression using exponential form.
5. 3 3 3 3 3
6. 2 2 2 2 2 2 2
7. 7 7 7 7 7
8. 10 10 10 10 10
9. 8 8 8 8 8 8
10. 5 5 5 5 5 5
28. 512
29. 18
Evaluate.
30. 36
11. 32
12. 23
31. 9
13. 24
14. 25
32. 49
15. 83
16. 35
33. 1296
17. 15
18. 44
34. 48
19. 52
20. 63
21. 94
22. 71
23. 103
24. 102
25. 106
26. 107
27. 2 43
28. (2 4)3
29. 2 32
30. (2 3)2
31. 5 22
32. (5 2)2
33. (3 2)4
34. 3 24
49
50
Chapter 0
■
The Arithmetic of Signed Numbers
35. 19
36. 2
37. 54
38. 12
39. 14
40. 14
41. 1
42. 6
43. 60
44. 1
45. 144
46. 6
47. 75
48. 40
49. 225
50. 1000
51. 34
52. 40
53. 21
54. 8
55. 4
56. 58
57. 40
58. 195
59. 256
60. 48
61. 196
62. 400
63. 147
64. 40
65. 21
66. 12
67. 25
68. 80
69. 96
70. 89
71. 15
72. 33
73. 9
74. 16
75. 6
76. 17
Evaluate each of the following expressions.
35. 7 2 6
36. 10 4 2
37. (7 2) 6
38. (10 4) 2
39. 12 8 4
40. 10 20 5
41. (12 8) 4
42. (10 20) 5
43. 8 7 2 2
44. 48 8 14 2
45. 8 (7 2) 2
46. 48 (8 4) 2
47. 3 52
48. 5 23
49. (3 5)2
50. (5 2)3
51. 4 32 2
52. 3 24 8
53. 7(23 5)
54. 4(32 7)
55. 3 24 26 2
56. 4 23 15 6
57. (2 4)2 8 3
58. (3 2)3 7 3
59. 4(2 6)2
60. 3(8 4)2
61. (4 2 6)2
62. (3 8 4)2
63. 3(4 3)2
64. 5(4 2)3
65. 3 4 32
66. 5 4 23
67. 4(2 3)2 125
68. 8 2(3 3)2
69. (4 2 3)2 25
70. 8 (2 3 3)2
71. 8 32 18 9
72. 14 3 9 28 7 2
73. 4 8 2 52
74. 12 8 4 2
75. 15 5 3 2 (2)3
76. 8 14 2 4 3
Section 0.5
■
Exponents and Order of Operations
51
Evaluate using your calculator. Round your answer to the nearest tenth.
77. (1.2)3 2.0736 2.4 1.6935 2.4896
78. (5.21 3.14 6.2154) 5.12 0.45625
77. 1.2
78. 1.5
79. 1.23 3.169 2.05194 (5.128 3.15 10.1742)
79. 7.8
80. 5.4
80. 4.56 (2.34)4 4.7896 6.93 27.5625 3.1269 (1.56)2
81. 25
82. 93
81. Population doubling. Over the last 2000 years, the Earth’s population has doubled approximately five times. Write the phrase “doubled five times” in exponential form.
83. 36 (4 2) 4
82. Volume of a cube. The volume of a cube with each edge of length 9 inches (in.)
is given by 9 9 9. Write the volume using exponential notation.
Many of the exercise sets in this text have a set of problems marked by the hurdler
logo shown here. These are particularly challenging exercises which either introduce
ideas that extend the material of the section or require you to generalize from what you
have learned.
83. Insert grouping symbols in the proper place so that the value of the expression
36 4 2 4 is 2.
84. Work with a small group of students.
Part 1: Write the numbers 1 through 25 on slips of paper and put the slips in a
pile, face down. Each of you randomly draws a slip of paper until you have drawn
five slips. Turn the papers over and write down the five numbers. Put the five papers back in the pile, shuffle, and then draw one more. This last number is the answer. The first five numbers are the problem. Your task is to arrange the first five
into a computation, using all you know about the order of operations, so that the
answer is the last number. Each number must be used and may be used only once.
If you cannot find a way to do this, pose it as a question to the whole class. Is this
guaranteed to work?
Part 2: Use your five numbers in a problem, each number being used and used
only once, for which the answer is 1. Try this nine more times with the numbers
2 through 10. You may find more than one way to do each of these. Surprising,
isn’t it?
Part 3: Be sure that when you successfully find a way to get the desired answer
using the five numbers, you can then write your steps using the correct order of
operations. Write your 10 problems and exchange them with another group to see
if they get these same answers when they do your problems.
Summary
10 20 50
1. , , 14 28 70
Exercises
■
0
This summary exercise set is provided to give you practice with each of the objectives
of the chapter. Each exercise is keyed to the appropriate chapter section. Your instructor
will give you guidelines on how to best use these exercises in your instructional setting.
9 15 30
2. , , 33 55 110
[0.1] In Exercises 1 to 3, write three fractional representations for each number.
8 16 24
3. , , 18 36 54
3
4. 8
1
5. 9
1
6. 6
2
7. 3
5
8. 6
13
9. 9
31
10. 36
7
11. 18
7
12. 54
13. 12
14. 8
15. 3
16. 20
17. 4
5
1. 7
3
2. 11
4
3. 9
24
4. Write the fraction in simplest form.
64
[0.1] In Exercises 5 to 12, perform the indicated operations.
18. 5
7
5
5. 15
21
10
9
6. 27
20
5
5
7. 12
8
7
14
8. 15
25
5
11
9. 6
18
5
7
10. 18
12
11
2
11. 18
9
11
5
12. 27
18
[0.2] In Exercises 13 to 20, complete the statements.
19. 16
20. 9
13. The absolute value of 12 is ________
14. The opposite of 8 is ________
15. 3 ________
16. (20) ________
17. 4 ________
18. (5) ________
19. The absolute value of 16 is ________.
20. The opposite of the absolute value of 9 is ________.
52
■
Summary Exercises
21. 22. 23. 24. 25. 8
26. 5
27. 11
1
28. 2
29. 4
30. 4
31. 4
32. 7
33. 5
34. 15
35. 5
36. 4
13
37. 2
38. 5
39. 1
40. 5
41. 0
42. 0
43. 36
44. 80
45. 15
3
46. 10
47. 16
48. 42
33. 15 20
34. 10 5
49. 360
50. 54
35. 2 (3)
36. 7 (3)
[0.2] Complete each of the following statements using the symbol , , or .
21. 3 ________ 1
22. 6 ________ 6
23. 8 ________ (3)
24. 5 ________ (5)
[0.3] In Exercises 25 to 32 add.
25. 15 (7)
26. 4 (9)
27. 8 (3)
5
4
28. 2
2
29. 3 (1)
30. 5 (6) (3)
31. 7 (4) 8 (7)
32. 6 9 9 (5)
[0.3] In Exercises 33 to 42, subtract.
23
3
37. 4
4
38. 3 2
39. 8 12 (5)
40. 6 7 (18)
41. 7 (4) 7 4
42. 9 (6) 8 (11)
[0.4] In Exercises 43 to 50, multiply.
43. (12)(3)
44. (10)(8)
45. (5)(3)
3
4
46. 8
5
47. (4)2
48. (2)(7)(3)
49. (6)(5)(4)(3)
50. (9)(2)(3)(1)
53
54
Chapter 0
■
The Arithmetic of Signed Numbers
51. 4
52. 121
53. 24
54. 36
55. 4
56. 11
57. Undefined
58. 25
7
59. 6
60. 0
61. 2
62. 12
63. 3 3 3
[0.4] In Exercises 51 to 54 use the distributive property to remove parentheses and
simplify.
51. 4(7 8)
52. 11(15 4)
53. 8(5 2)
54. 4(3 6)
[0.4] In Exercises 55 to 62, divide.
64. 5 5 5 5
65. 2 2 2 2 2 2
66. 4 4 4 4 4
67. 3
68. 75
69. 80
70. 400
55. 48 (12)
33
56. 3
57. 9 0
58. 75 (3)
7
2
59. 9
3
60. 0 (12)
61. 8 4
62. (12) (1)
71. 41
72. 25
73. 20
[0.5] Write each of these in expanded form.
74. 16
75. 324
63. 33
64. 54
76. 27
65. 26
66. 45
77. 11
78. 169
[0.5] Evaluate each of the following expressions.
67. 18 3 5
68. (18 3) 5
69. 5 42
70. (5 4)2
71. 5 32 4
72. 5 ( 3 2 4 )
73. 5(4 2)2
74. 5 4 22
75. (5 4 2)2
76. 3(5 2)2
77. 3 5 22
78. (3 5 2)2