2.5: Order of Operations

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Aug 30, 2024
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- 2.4: Multiplication and Division of Integers
- 2.6: Solving Equations

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For convenience, we repeat the “Rules Guiding Order of Operations” first introduced in Section 1.5.

Rules Guiding Order of Operations

When evaluating expressions, proceed in the following order.

Evaluate expressions contained in grouping symbols first. If grouping symbols are nested, evaluate the expression in the innermost pair of grouping symbols first.
Evaluate all exponents that appear in the expression.
Perform all multiplications and divisions in the order that they appear in the expression, moving left to right.
Perform all additions and subtractions in the order that they appear in the expression, moving left to right.

Let’s look at a number of examples that require the use of these rules.

Example 1

Simplify: (a) (−3)² and (b) −3²

Solution

Recall that for any integer a, we have (−1)a = −a. Because negating is equivalent to multiplying by −1, the “Rules Guiding Order of Operations” require that we address grouping symbols and exponents before negation.

a) Because of the grouping symbols, we negate first, then square. That is,

$\begin{aligned} (−3)^2 = (−3)(−3) \\ & = 9. \end{aligned}\nonumber$

b) There are no grouping symbols in this example. Thus, we must square first, then negate. That is,

$\begin{aligned} −3^2 = −(3 \cdot 3) \\ = −9. \end{aligned}\nonumber$

Exercise

Simplify: −2².

Answer: −4

Example 2

Simplify: −2 − 3(5 − 7).

Solution

Grouping symbols first, then multiplication, then subtraction.

$\begin{aligned} -2-3(5-7)=-2-3(-2) ~ & \textcolor{red}{ \text{ Perform subtraction within parentheses.}} \\ =2 -2-(-6) ~ & \textcolor{red}{ \text{ Multiply: } 3(-2)=-6.} \\ = -2+6 ~ & \textcolor{red}{ \text{ Add the opposite.}} \\ =4 \end{aligned}\nonumber$

Exercise

Simplify: −3 − 2(6 − 8).

Answer: 1

Example 3

Simplify: −2(2 − 4)² − 3(3 − 5)³.

Solution

Grouping symbols first, then multiplication, and subtraction, in that order.

\[ \begin{aligned} -2(2-4)^2 -3(3-5)^3 = -2(-2)^2 -3(-2)^3 ~ & \textcolor{red}{ \text{ Perform subtraction within parentheses first.} \\ =2 (4) -3(-8) ~ & \textcolor{red}{ \text{ Exponents are next.}} \\ =-8-(-24) ~ & \textcolor{red}{ \text{ Multiplications are next.}} \\ =-8+24 ~ & \textcolor{red}{ \text{ Add the opposite.}} \\ =16 \end{aligned}\nonumber \]

Exercise

Simplify: −2(5 − 6)³ − 3(5 − 7)²

Answer: -10

Example 4

Simplify: −24 ÷ 8(−3).

Solution

Division has no preference over multiplication, or vice versa. Divisions and multiplications must be performed in the order that they occur, moving left to right.

$\begin{aligned} -24 \div 8(-3) = -3(-3) ~ & \textcolor{red}{ \text{ Division first: } -24 \div 8 = -3.} \\ =9 \end{aligned}\nonumber$

Note that if you multiply first, which would be incorrect, you would get a completely different answer.

Exercise

Simplify: −48 ÷ 6(−2).

Answer: 16

Example 5

Simplify: (−2)(−3)(−2)³.

Solution

Exponents first, then multiplication in the order that it occurs, moving left to right.

$\begin{aligned} (-2)(-3)(-2)^3 = (-2)(-3)(-8) ~ & \textcolor{red}{ \text{ Exponent first: } (-2)^3 = -8.} \\ =6(-8) ~ & \textcolor{red}{ \text{ Multiply from left to right: } (-2)(-3) = 6.} \\ =-48 \end{aligned}\nonumber$

You try it!

Simplify: (−4)(−2)²(−1)³.

Answer: 16

Evaluating Fractions

If a fraction bar is present, evaluate the numerator and denominator separately according to the “Rules Guiding Order of Operations,” then perform the division in the final step.

Example 6

Simplify:

$\frac{-5-5(2-4)^3}{-22 - 3(-5)}\nonumber$

Solution

Evaluate numerator and denominator separately, then divide.

$\begin{aligned} \frac{-5-5(2-4)^3}{-22-3(-5)} = \frac{-5-5(-2)^3}{-22-(-15)} ~ & \begin{array}{l} \textcolor{red}{ \text{ Numerator: parentheses first.}} \\ \textcolor{red}{ \text{ Denominator: multiply } 3(-5)=-15.} \end{array} \\ = \frac{-5-5(-8)}{-22+15} ~ & \begin{array}{l} \textcolor{red}{ \text{ Numerator: exponent } (-2)^3 = -8.} \\ \textcolor{red}{ \text{ Denominator: add the opposite.}} \end{array} \\ = \frac{-5-(-40)}{-7} & \begin{array}{l} \textcolor{red}{ \text{ Numerator: multiply } 5(-8) = -40.} \\ \textcolor{red}{ \text{ Denominator: add } -22 + 15 = -7.} \end{array} \\ = \frac{-5+40}{-7} ~ & \textcolor{red}{ \text{ Numerator: add the opposite.}} \\ = \frac{35}{-7} ~ & \textcolor{red}{ \text{ Numerator: } -5 + 40 = 35.} \\ = -5 ~ & \textcolor{red}{ \text{ Divide: } 35/-7 = -5.} \end{aligned}\nonumber$

Exercise

Simplify:

$\frac{6-2(-6)}{-2-(-2)^2}\nonumber$

Answer: -3

Absolute Value

Absolute value calculates the magnitude of the vector associated with an integer, which is equal to the distance between the number and the origin (zero) on the number line. Thus, for example, |4| = 4 and | − 5| = 5.

But absolute value bars also act as grouping symbols, and according to the “Rules Guiding Order of Operations,” you should evaluate the expression inside a pair of grouping symbols first.

Example 7

Simplify: (a) −(−3) and (b) −| − 3|.

Solution

There is a huge difference between simple grouping symbols and absolute value.

a) This is a case of −(−a) = a. Thus, −(−3) = 3.

b) This case is much different. The absolute value of −3 is 3, and then the negative of that is −3. In symbols,

−| − 3| = −3

Exercise

Simplify: −|−8|.

Answer: −8

Example 8

Simplify: −3 − 2|5 − 7|.

Solution

Evaluate the expression inside the absolute value bars first. Then multiply, then subtract.

$\begin{aligned} -3-2|5-7|=-3-2|-2| ~ & \end{aligned}\nonumber$

Exercise

Simplify: −2 − 4|6 − 8|.

Answer: −10

Exercises

In Exercises 1-40, compute the exact value of the given expression.

1. $7 - \frac{-14}{2}$

2. $-2 - \frac{-16}{4}$

3. $-7 - \frac{-18}{9}$

4. $-6 - \frac{-7}{7}$

5. −54

6. −33

7. 9 − 1(−7)

8. 85 − 8(9)

9. −63

10. −35

11. 3 + 9(4)

12. 6 + 7(−1)

13. 10 − 72 ÷ 6 · 3+8

14. 8 − 120 ÷ 5 · 6+7

15. $6 + \frac{14}{2}$

16. $16 + \frac{8}{2}$

17. −34

18. −22

19. 3 − 24 ÷ 4 · 3+4

20. 4 − 40 ÷ 5 · 4+9

21. 64 ÷ 4 · 4

22. 18 ÷ 6 · 1

23. −2 − 3(−5)

24. 64 − 7(7)

25. 15 ÷ 1 · 3

26. 30 ÷ 3 · 5

27. 8 + 12 ÷ 6 · 1 − 5

28. 9 + 16 ÷ 2 · 4 − 9

29. 32 ÷ 4 · 4

30. 72 ÷ 4 · 6

31. $-11 + \frac{16}{16}$

32. $4 + \frac{-20}{4}$

33. −52

34. −43

35. 10 + 12(−5)

36. 4 + 12(4)

37. 2+6 ÷ 1 · 6 − 1

38. 1 + 12 ÷ 2 · 2 − 6

39. 40 ÷ 5 · 4

40. 30 ÷ 6 · 5

In Exercises 41-80, simplify the given expression.

41. −11 + | − 1 − (−6)²|

42. 13 + | − 21 − (−4)²|

43. |0(−4)| − 4(−4)

44. |10(−3)| − 3(−1)

45. (2 + 3 · 4) − 8

46. (11 + 5 · 2) − 10

47. (8 − 1 · 12) + 4

48. (9 − 6 · 1) + 3

49. (6 + 10 · 4) − 6

50. (8 + 7 · 6) − 12

51. 10 + (6 − 4)³ − 3

52. 5 + (12 − 7)² − 6

53. (6 − 8)² − (4 − 7)³

54. (3 − 8)² − (4 − 9)³

55. |0(−10)| + 4(−4)

56. |12(−5)| + 7(−5)

57. |8(−1)| − 8(−7)

58. |6(−11)| − 7(−1)

59. 3 + (3 − 8)² − 7

60. 9 + (8 − 3)³ − 6

61. (4 − 2)² − (7 − 2)³

62. (1 − 4)² − (3 − 6)³

63. 8 −|− 25 − (−4)²|

64. 20 −|− 22 − 4²|

65. −4 − |30 − (−5)²|

66. −8 −|− 11 − (−6)²|

67. (8 − 7)² − (2 − 6)³

68. (2 − 7)² − (4 − 7)³

69. 4 − (3 − 6)³ + 4

70. 6 − (7 − 8)³ + 2

71. −3 + | − 22 − 5²|

72. 12 + |23 − (−6)²|

73. (3 − 4 · 1) + 6

74. (12 − 1 · 6) + 4

75. 1 − (1 − 5)² + 11

76. 9 − (3 − 1)³ + 10

77. (2 − 6)² − (8 − 6)³

78. (2 − 7)² − (2 − 4)³

79. |9(−3)| + 12(−2)

80. |0(−3)| + 9(−7)

In Exercises 81-104, simplify the given expression.

81. $\frac{4(-10) -5}{-9}$

82. $\frac{-4 \cdot 6 - (-8)}{-4}$

83. $\frac{10^2 - 4^2}{2 \cdot 6 - 10}$

84. $\frac{3^2 - 9^2}{2 \cdot 7 - 5}$

85. $\frac{3^2 + 6^2}{5 - 1 \cdot 8}$

86. $\frac{10^2 + 4^2}{1 - 6 \cdot 5}$

87. $\frac{-8-4}{7 - 13}$

88. $\frac{13-1}{8-4}$

89. $\frac{2^2 + 6^2}{11 - 4 \cdot 4}$

90. $\frac{7^2 + 3^2}{10 - 8 \cdot 1}$

91. $\frac{1^2 - 5^2}{9 \cdot 1 - 5}$

92. $\frac{5^2 - 7^2}{2 \cdot 2 - 12}$

93. $\frac{4^2 - 8^2}{6 \cdot 3 - 2}$

94. $\frac{7^2 - 6^2}{6 \cdot 3 - 5}$

95. $\frac{10^2 + 2^2}{10-2 \cdot 7}$

96. $\frac{2^2 + 10^2}{10 - 2 \cdot 7}$

97. $\frac{16-(-2)}{19-1}$

98. $\frac{-8-20}{-15-(-17}$

99. $\frac{15 -(-15)}{13-(-17)}$

100. $\frac{7-(-9)}{-1-1}$

101. $\frac{4 \cdot 5 - (-19)}{3}$

102. $\frac{10 \cdot 7 - (-11)}{-3}$

103. $\frac{-6 \cdot 9 -(-4)}{2}$

104. $\frac{-6 \cdot 2 - 10}{-11}$

Answers

1. 14

3. −5

5. −625

7. 16

9. −216

11. 39

13. −18

15. 13

17. −81

19. −11

21. 64

23. 13

25. 45

27. 5

29. 32

31. −10

33. −25

35. −50

37. 37

39. 32

41. 26

43. 16

45. 6

47. 0

49. 40

51. 15

53. 31

55. −16

57. 64

59. 21

61. −121

63. −33

65. −9

67. 65

69. 35

71. 44

73. 5

75. −4

77. 8

79. 3

81. 5

83. 42

85. −15

87. 2

89. −8

91. −6

93. −3

95. −8

97. 1

99. 1

101. 13

103. −25

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Example 6

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Example 7

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Example 8

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Rules Guiding Order of Operations

Example 1

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Example 2

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Example 3

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Example 4

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Example 5

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You try it!

Evaluating Fractions

Example 6

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Exercise

Absolute Value

Example 7

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Exercise

Example 8

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Exercise

Exercises

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