OpenStax-CNX module: m34872
1
Exponents, Roots, Factorization of
Whole Numbers: Grouping Symbols
∗
and the Order of Operations
Wade Ellis
Denny Burzynski
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 3.0
†
Abstract
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr.
module discusses grouping symbols and the order of operations.
This
By the end of the module students
should be able to understand the use of grouping symbols, understand and be able to use the order of
operations and use the calculator to determine the value of a numerical expression.
1 Section Overview
•
•
•
•
Grouping Symbols
Multiple Grouping Symbols
The Order of Operations
Calculators
2 Grouping Symbols
Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations are
to be grouped together and considered as one number. The grouping symbols commonly used in mathematics
are the following:
( ), [ ], { },
Parentheses:
Brackets: [ ]
Braces: { }
Bar:
()
In a computation in which more than one operation is involved, grouping symbols indicate which operation
to perform rst. If possible, we perform operations inside grouping symbols rst.
∗ Version
1.2: Aug 18, 2010 8:25 pm -0500
† http://creativecommons.org/licenses/by/3.0/
http://cnx.org/content/m34872/1.2/
OpenStax-CNX module: m34872
2
2.1 Sample Set A
If possible, determine the value of each of the following.
Example 1
9 + (3 · 8)
Since 3 and 8 are within parentheses, they are to be combined rst.
9 + (3 · 8)
= 9 + 24
= 33
Thus,
9 + (3 · 8) = 33
Example 2
(10 ÷ 0) · 6
Since 10 ÷ 0 is undened, this operation is meaningless, and we attach no value to it. We write,
"undened."
2.2 Practice Set A
If possible, determine the value of each of the following.
Exercise 1
(Solution on p. 12.)
Exercise 2
(Solution on p. 12.)
16 − (3 · 2)
5 + (7 · 9)
Exercise 3
(Solution on p. 12.)
(4 + 8) · 2
Exercise 4
(Solution on p. 12.)
Exercise 5
(Solution on p. 12.)
Exercise 6
(Solution on p. 12.)
28 ÷ (18 − 11)
(33 ÷ 3) − 11
4 + (0 ÷ 0)
3 Multiple Grouping Symbols
When a set of grouping symbols occurs inside another set of grouping symbols, we perform the operations
within the innermost set rst.
3.1 Sample Set B
Determine the value of each of the following.
Example 3
2 + (8 · 3) − (5 + 6)
Combine 8 and 3 rst, then combine 5 and 6.
2 + 24 − 11 Now combine left to right.
26 − 11
15
http://cnx.org/content/m34872/1.2/
OpenStax-CNX module: m34872
3
Example 4
10 + [30 − (2 · 9)]
Combine 2 and 9 since they occur in the innermost set of parentheses.
10 + [30 − 18] Now combine 30 and 18.
10 + 12
22
3.2 Practice Set B
Determine the value of each of the following.
Exercise 7
(Solution on p. 12.)
Exercise 8
(Solution on p. 12.)
Exercise 9
(Solution on p. 12.)
Exercise 10
(Solution on p. 12.)
Exercise 11
(Solution on p. 12.)
(17 + 8) + (9 + 20)
(55 − 6) − (13 · 2)
23 + (12 ÷ 4) − (11 · 2)
86 + [14 ÷ (10 − 8)]
31 + {9 + [1 + (35 − 2)]}
Exercise 12
{6 − [24 ÷ (4 · 2)]}
3
(Solution on p. 12.)
4 The Order of Operations
Sometimes there are no grouping symbols indicating which operations to perform rst. For example, suppose
we wish to nd the value of 3 + 5 · 2. We could do either of two things:
Add 3 and 5, then multiply this sum by 2.
3+5·2
= 8 · 2
= 16
Multiply 5 and 2, then add 3 to this product.
3+5·2
= 3 + 10
= 13
We now have two values for one number. To determine the correct value, we must use the accepted order
of operations.
Order of Operations
1. Perform all operations inside grouping symbols, beginning with the innermost set, in the order 2, 3, 4
described below,
2. Perform all exponential and root operations.
3. Perform all multiplications and divisions, moving left to right.
4. Perform all additions and subtractions, moving left to right.
http://cnx.org/content/m34872/1.2/
OpenStax-CNX module: m34872
4
4.1 Sample Set C
Determine the value of each of the following.
Example 5
21 + 3 · 12
Multiply rst.
21 + 36
Add.
57
Example 6
(15 − 8) + 5 · (6 + 4) . Simplify inside parentheses rst.
7 + 5 · 10
Multiply.
7 + 50
Add.
57
Example 7
63 − (4 + 6 · 3) + 76 − 4
Simplify rst within the parenthesis by multiplying, then adding.
63 − (4 + 18) + 76 − 4
63 − 22 + 76 − 4
Now perform the additions and subtractions, moving left to right.
41 + 76 − 4
Add 41 and 76: 41 + 76 = 117.
Subtract 4 from 117:
117 − 4
117 − 4 = 113.
113
Example 8
7 · 6 − 42 + 15
Evaluate the exponential forms, moving left to right.
7 · 6 − 16 + 1
Multiply 7 and 6: 7 · 6 = 42
42 − 16 + 1
Subtract 16 from 42: 42 − 16 = 26
26 + 1
Add 26 and 1: 26 + 1 = 27
27
Example 9
6 · 32 + 2 2 + 4 2
Evaluate the exponential forms in the parentheses: 32 = 9 and 22 = 4
6 · (9 + 4) + 42
Add the 9 and 4 in the parentheses: 9 + 4 = 13
6 · (13) + 4
Evaluate the exponential form: 42 = 16
2
6 · (13) + 16
Multiply 6 and 13: 6 · 13 = 78
78 + 16
Add 78 and 16: 78 + 16 = 94
94
http://cnx.org/content/m34872/1.2/
OpenStax-CNX module: m34872
5
Example 10
62 +22
42 +6·22
36+4
16+6·4
Recall that the bar is a grouping symbol.
13 +82
+
102 −19·5
+
1+64
100−19·5
+
1+64
100−95
36+4
16+24
40
40
+
+2
2
2
The fraction 462 +6·2
÷ 42 + 6 · 22
2 is equivalent to 6 + 2
2
2
65
5
1 + 13
14
4.2 Practice Set C
Determine the value of each of the following.
Exercise 13
(Solution on p. 12.)
Exercise 14
(Solution on p. 12.)
Exercise 15
(Solution on p. 12.)
Exercise 16
(Solution on p. 12.)
8 + (32 − 7)
(34 + 18 − 2 · 3) + 11
8 (10) + 4 (2 + 3) − (20 + 3 · 15 + 40 − 5)
5 · 8 + 42 − 22
Exercise 17
(Solution on p. 12.)
4 62 − 33 ÷ 42 − 4
Exercise 18
(Solution on p. 12.)
(8 + 9 · 3) ÷ 7 + 5 · (8 ÷ 4 + 7 + 3 · 5)
Exercise 19
33 +23
62 −29
+5
(Solution on p. 12.)
82 +24
72 −32
÷
8·3+18
23 −3
5 Calculators
Using a calculator is helpful for simplifying computations that involve large numbers.
5.1 Sample Set D
Use a calculator to determine each value.
Example 11
9, 842 + 56 · 85
http://cnx.org/content/m34872/1.2/
OpenStax-CNX module: m34872
6
Key
Perform the multiplication rst.
Now perform the addition.
Display Reads
Type
56
56
Press
×
56
Type
85
85
Press
+
4760
Type
9842
9842
Press
=
14602
Table 1
The display now reads 14,602.
Example 12
42 (27 + 18) + 105 (810 ÷ 18)
Key
Operate inside the parentheses
Multiply by 42.
Display Reads
Type
27
27
Press
+
27
Type
18
18
Press
=
45
Press
×
45
Type
42
42
Press
=
1890
Table 2
Place this result into memory by pressing the memory key.
Key
Now operate in the other parentheses.
Now multiply by 105.
We are now ready to add these two quantities together.
Display Reads
Type
810
810
Press
÷
810
Type
18
18
Press
=
45
Press
×
45
Type
105
105
Press
=
4725
Press
+
4725
Press the memory recall key.
1890
Press
Table 3
http://cnx.org/content/m34872/1.2/
=
6615
OpenStax-CNX module: m34872
7
Thus, 42 (27 + 18) + 105 (810 ÷ 18) = 6, 615
Example 13
164 + 373
Nonscientic Calculators
Key
Display Reads
Type
16
16
Press
×
16
Type
16
16
Press
×
256
Type
16
16
Press
×
4096
Type
16
16
Press
=
65536
Type
37
37
Press
×
37
Type
37
37
Press
×
1396
Type
37
37
Press
×
50653
Press
+
50653
Press the memory key
Press memory recall key
Press
65536
=
116189
Table 4
Calculators with y x Key
Key
http://cnx.org/content/m34872/1.2/
Display Reads
Type
16
16
Press
y
x
16
Type
4
4
Press
=
4096
Press
+
4096
Type
37
37
Press
y
x
37
Type
3
3
Press
=
116189
OpenStax-CNX module: m34872
8
Table 5
Thus, 164 + 373 = 116, 189
We can certainly see that the more powerful calculator simplies computations.
Example 14
Nonscientic calculators are unable to handle calculations involving very large numbers.
85612 · 21065
Key
Display Reads
Type
85612
85612
Press
×
85612
Type
21065
21065
Press
=
Table 6
This number is too big for the display of some calculators and we'll probably get some kind of
error message. On some scientic calculators such large numbers are coped with by placing them
in a form called "scientic notation." Others can do the multiplication directly. (1803416780)
5.2 Practice Set D
Use a calculator to nd each value.
Exercise 20
(Solution on p. 12.)
Exercise 21
(Solution on p. 12.)
Exercise 22
(Solution on p. 12.)
Exercise 23
(Solution on p. 12.)
9, 285 + 86 (49)
55 (84 − 26) + 120 (512 − 488)
1063 − 174
6, 0533
6 Exercises
For the following problems, nd each value. Check each result with a calculator.
Exercise 24
(Solution on p. 12.)
2 + 3 · (8)
Exercise 25
18 + 7 · (4 − 1)
Exercise 26
3 + 8 · (6 − 2) + 11
(Solution on p. 12.)
Exercise 27
1 − 5 · (8 − 8)
Exercise 28
37 − 1 · 62
http://cnx.org/content/m34872/1.2/
(Solution on p. 13.)
OpenStax-CNX module: m34872
9
Exercise 29
98 ÷ 2 ÷ 72
Exercise 30
(Solution on p. 13.)
42 − 2 · 4 − 23
Exercise
31
√
9 + 14
Exercise
√
√32
(Solution on p. 13.)
100 + 81 − 42
Exercise
33
√
3
8+8−2·5
Exercise
34
√
4
(Solution on p. 13.)
16 − 1 + 52
Exercise 35
61 − 22 + 4 [3 · (10) + 11]
Exercise 36
121 − 4 · [(4) · (5) − 12] +
(Solution on p. 13.)
16
2
Exercise 37
(1+16)−3
7
+ 5 · (12)
Exercise 38
8·(6+20)
8
+
(Solution on p. 13.)
3·(6+16)
22
Exercise 39
10 · [8 + 2 · (6 + 7)]
Exercise 40
(Solution on p. 13.)
21 ÷ 7 ÷ 3
Exercise 41
102 · 3 ÷ 52 · 3 − 2 · 3
Exercise 42
(Solution on p. 13.)
85 ÷ 5 · 5 − 85
Exercise 43
51
17
+7−2·5·
12
3
Exercise 44
22 · 3 + 23 · (6 − 2) − (3 + 17) + 11 (6)
(Solution on p. 13.)
Exercise 45
26 − 2 · { 6+1320 }
Exercise 46
(Solution on p. 13.)
2 · {(7 + 7) + 6 · [4 · (8 + 2)]}
Exercise 47
0 + 10 (0) + 15 · {4 · 3 + 1}
Exercise 48
18 +
(Solution on p. 13.)
7+2
9
Exercise 49
(4 + 7) · (8 − 3)
Exercise 50
(6 + 8) · (5 + 2 − 4)
http://cnx.org/content/m34872/1.2/
(Solution on p. 13.)
OpenStax-CNX module: m34872
10
Exercise 51
(21 − 3) · (6 − 1) · (7) + 4 (6 + 3)
Exercise 52
(10 + 5) · (10 + 5) − 4 · (60 − 4)
Exercise 53
6 · {2 · 8 + 3} − (5) · (2) +
8
4
(Solution on p. 13.)
+ (1 + 8) · (1 + 11)
Exercise 54
(Solution on p. 13.)
25 + 3 · (8 + 1)
Exercise 55
34 + 24 · (1 + 5)
Exercise 56
16 + 08 + 52 · (2 + 8)
(Solution on p. 13.)
3
Exercise 57
(7) · (16) − 34 + 22 · 17 + 32
Exercise 58
(Solution on p. 13.)
23 −7
52
Exercise
59
2
(1+6) +2
3·6+1
Exercise 60
62 −1
23 −3
+
(Solution on p. 13.)
43 +2·3
2·5
Exercise 61
5(82 −9·6)
25 −7
+
72 −42
24 −5
Exercise 62
(2+1)3 +23 +110
62
Exercise 63
63 −2·102
22
+
(Solution on p. 13.)
−
15
2
−[2·5]2
5·52
(23 +72 )
2(19)−33
18
Exercise 64
√ 2 · {6 + 102 − 6 25 }
Exercise 65√
(Solution on p. 13.)
√ 181 − 3 · 2 36 + 3 3 64
Exercise
66
√
√
(Solution on p. 13.)
2·( 81− 3 125)
42 −10+22
6.1 Exercises for Review
Exercise 67
( here1 ) The fact that 0 + any whole number = that particular whole number is an example of
which property of addition?
Exercise 68
( here2 ) Find the product. 4, 271 × 630.
(Solution on p. 13.)
1 "Addition and Subtraction of Whole Numbers: Properties of Addition" <http://cnx.org/content/m34802/latest/>
2 "Multiplication and Division of Whole Numbers: Multiplication of Whole Numbers"
<http://cnx.org/content/m34863/latest/>
http://cnx.org/content/m34872/1.2/
OpenStax-CNX module: m34872
11
Exercise 69
( here3 ) In the statement 27 ÷ 3 = 9, what name is given to the result 9?
Exercise 70
( here4 ) What number is the multiplicative identity?
(Solution on p. 13.)
Exercise 71
( here5 ) Find the value of 24 .
3 "Multiplication and Division of Whole Numbers: Concepts of Division of Whole Numbers"
<http://cnx.org/content/m34864/latest/>
4 "Multiplication and Division of Whole Numbers: Summary of Key Concepts" <http://cnx.org/content/m34868/latest/>
5 "Multiplication and Division of Whole Numbers: Summary of Key Concepts" <http://cnx.org/content/m34868/latest/>
http://cnx.org/content/m34872/1.2/
OpenStax-CNX module: m34872
12
Solutions to Exercises in this Module
Solution to Exercise (p. 2)
10
Solution to Exercise (p. 2)
68
Solution to Exercise (p. 2)
24
Solution to Exercise (p. 2)
4
Solution to Exercise (p. 2)
0
Solution to Exercise (p. 2)
not possible (indeterminant)
Solution to Exercise (p. 3)
54
Solution to Exercise (p. 3)
23
Solution to Exercise (p. 3)
4
Solution to Exercise (p. 3)
93
Solution to Exercise (p. 3)
74
Solution to Exercise (p. 3)
27
Solution to Exercise (p. 5)
33
Solution to Exercise (p. 5)
57
Solution to Exercise (p. 5)
0
Solution to Exercise (p. 5)
52
Solution to Exercise (p. 5)
3
Solution to Exercise (p. 5)
125
Solution to Exercise (p. 5)
7
Solution to Exercise (p. 8)
13,499
Solution to Exercise (p. 8)
6,070
Solution to Exercise (p. 8)
1,107,495
Solution to Exercise (p. 8)
This number is too big for a nonscientic calculator. A scientic calculator will probably give you
2.217747109 × 1011
Solution to Exercise (p. 8)
26
http://cnx.org/content/m34872/1.2/
OpenStax-CNX module: m34872
Solution to Exercise (p. 8)
46
Solution to Exercise (p. 8)
1
Solution to Exercise (p. 9)
0
Solution to Exercise (p. 9)
3
Solution to Exercise (p. 9)
26
Solution to Exercise (p. 9)
97
Solution to Exercise (p. 9)
29
Solution to Exercise (p. 9)
1
Solution to Exercise (p. 9)
0
Solution to Exercise (p. 9)
90
Solution to Exercise (p. 9)
508
Solution to Exercise (p. 9)
19
Solution to Exercise (p. 9)
144
Solution to Exercise (p. 10)
1
Solution to Exercise (p. 10)
52
Solution to Exercise (p. 10)
25,001
Solution to Exercise (p. 10)
1
25
Solution to Exercise (p. 10)
14
Solution to Exercise (p. 10)
0
Solution to Exercise (p. 10)
152
Solution to Exercise (p. 10)
4
5
Solution to Exercise (p. 10)
2,690,730
Solution to Exercise (p. 11)
1
http://cnx.org/content/m34872/1.2/
13