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2.4: Order of Operations

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    The order in which we evaluate expressions can be ambiguous. Take for example, the expression 4 + 3 · 2. If we do the addition first, then

    4+3 · 2=7 · 2

    = 14.

    On the other hand, if we do the multiplication first, then

    4+3 · 2=4+6

    = 10.

    So, what are we to do? Of course, grouping symbols can remove the ambiguity

    Grouping Symbols

    Parentheses, brackets, or curly braces can be used to group parts of an expression. Each of the following are equivalent:

    (4 + 3) · 2 or

    [4 + 3] · 2 or

    {4+3} · 2

    In each case, the rule is “evaluate the expression inside the grouping symbols first.” If grouping symbols are nested, evaluate the expression in the innermost pair of grouping symbols first.

    For example,

    (4 + 3) · 2

    7 · 2

     14

    Note how the expression contained in the parentheses was evaluated first. Another way to avoid ambiguities in evaluating expressions is to establish an order in which operations should be performed. The following guidelines should always be strictly enforced when evaluating expressions.

    Rules Guiding Order of Operations

    When evaluating expressions, proceed in the following order.

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

    There are many ways in which Order of Operations is taught across the world. In the United States two of the most common mnemonic device is 
    "PEMDAS" - Parentheses, Exponents, Multiplication & Division, Addition & Subtraction. There is also the phrase "Please excuse my dear Aunt Sally" Which harbors the same meaning, just different words. Ask your classmates how they were taught this. You will find students have a broad range of systems to remember this rule.

     

    Example 1

    Evaluate 4 + 3 · 2.

    Solution

    Because of the established Rules Guiding Order of Operations, this expression is no longer ambiguous. There are no grouping symbols or exponents, so we immediately go to rule three, evaluate all multiplications and divisions in the order that they appear, moving left to right. After that we invoke rule four, performing all additions and subtractions in the order that they appear, moving left to right.

    \[ \begin{aligned} 4+3 \cdot 2 \\ 4+6 \\ 10 \end{aligned}\nonumber \]

    Thus, 4 + 3 · 2 simplifies to 10.

    Exercise

    Simplify: 8 + 2 · 5.

    Answer

    18

    Example 2

    Evaluate -18 − 2 + 3.

    Solution

    Follow the Rules Guiding Order of Operations. Addition has no precedence over subtraction, nor does subtraction have precedence over addition. We are to perform additions and subtractions as they occur, moving left to right.

    \[ \begin{aligned} -18 − 2 + 3 &\textcolor{red}{ \text{ Subtract: -18 − 2 gives -20.}} \\ -20 + 3 & \textcolor{red}{ \text{ Add: -20 + 3 gives -17.}} \\  -17 & \textcolor{red}{ \text{ Most Simplified Form }} \end{aligned}\nonumber \]

    Thus, -18 − 2 + 3 simplifies to -17.

    Exercise

    Simplify: 17 − 8 + 2.

    Answer

    11

    Example 3

    Evaluate 54 ÷ 9 · 2.

    Solution

    Follow the Rules Guiding Order of Operations. Division has no precedence over multiplication, nor does multiplication have precedence over division. We are to perform divisions and multiplications as they occur, moving left to right.

    \[ \begin{aligned} 54 \div 9 \cdot 2 & \textcolor{red}{ \text{ Divide: 54 by 9 simplifies to 6}} \\ 6 \cdot 2 & \textcolor{red}{ \text{ Multiply: 6 } \cdot \text{ 2 simplifies to 12. }}\\ 12 & \textcolor{red}{ \text{ Most Simplified Form}} \end{aligned}\nonumber \]

    Thus, 54 ÷ 9 · 2 simplifies to 12.

    Exercise

    Simplify: 72 ÷ 9 · 2.

    Answer

    16

    Example 4

    Evaluate 2 · 32 − 12.

    Solution

    Follow the Rules Guiding Order of Operations, exponents first, then multiplication, then subtraction.

    \[ \begin{aligned} 2 \cdot 3^2 - 12 & \textcolor{red}{ \text{ There are no group symbols, so evaluate the exponent: } 3^2 \text{ simplifies to } 9. } \\ 2 \cdot 9 - 12 & \textcolor{red}{ \text{ Multiply: } 2 \cdot 9 \text{ simplifies to } 18. } \\ 18 - 12 & \textcolor{red}{ \text{ Subtract: } 18-12 \text{ simplifies to } 6. } \\ 6 & \textcolor{red}{ \text{ Most Simplified Form }} \\ \end{aligned}\nonumber \]

    Thus, 2 · 32 − 12 simplifies to 6.

    Exercise

    Simplify: 14 + 3 · 42

    Answer

    62

    Example 5

    Evaluate 12 + 2(3 + 2 · 5)2.

    Solution

    Follow the Rules Guiding Order of Operations, evaluate the expression inside the parentheses first, then exponents, then multiplication, then addition.

    \[ \begin{aligned} 12 + 2(3 + 5 \cdot 5 )^2 ~ & \textcolor{red}{ \text{ Multiply inside parentheses: 2 } \cdot 5 \text{ to get } 10.} \\ 12 + 2(3 + 10)^2 ~ & \textcolor{red}{ \text{ Add inside parentheses/grouping symbols: } 3 + 10 \text{ to get }  13.} \\ 12 + 2(13)^2 ~ & \textcolor{red}{ \text{ Exponents are next: } (13)^2 \text{ to get }  169.} \\ 12 + 2(169)~ & \textcolor{red}{ \text{ Multiplication is next: } 2(169) \text{ to get }  338.} \\ 12 + 338 ~ & \textcolor{red}{ \text{ Finally add: } 12 + 338 \text{ to get }  350.} \\ 350 ~ & \textcolor{red}{ \text{ Most Simplified Form }} \end{aligned}\nonumber \]

    Thus, 12 + 2(3 + 2 · 5) 2 simplifies to 350.

    Exercise

    Simplify: 3(2 + 3 · 4)2 − 11.

    Answer

    577

    Example 6

    Evaluate 2{2 + 2[2 + 2]}.

    Solution

    When grouping symbols are nested, evaluate the expression between the pair of innermost grouping symbols first.

    \[ \begin{aligned} 2( 2 + 2[2 + 2]) ~ & \textcolor{red}{ \text{ Innermost grouping first: } 2 + 2 \text { to get } 4.}\\ 2(2 + 2[4]) ~ & \textcolor{red}{ \text{ Multiply next: } 2[4] \text { to get } 8.}  \\ 2(2+8) ~ & \textcolor{red}{ \text{ Add inside braces: } 2 + 8 \text { to get } 10.} \\ 2(10) ~ & \textcolor{red}{ \text{ Multiply: } 2(10) \text { to get } 20}\\ 20 ~ & \textcolor{red}{ \text{Most Simplified Form }} \end{aligned}\nonumber \]

    Thus, 2(2 + 2[2 + 2]) simplifies to 20.

    Exercise

    Simplify: 2{3 + 2[3 + 2]}.

    Answer

    26

    Fraction Bars

    Consider the expression

    \[ \frac{6^{2}+8^{2}}{(2+3)^{2}}\nonumber \]

    Because a fraction bar means division, the above expression is equivalent to

    \[\left(6^{2}+8^{2}\right) \div(2+3)^{2}\nonumber \]

    The position of the grouping symbols signals how we should proceed. We should simplify the numerator, then the denominator, then divide.

    Fractional Expressions

    If a fractional expression is present, evaluate the numerator and denominator first, then divide.

    Example 7

    Evaluate the expression

    \[ \frac{(6)^{2}+8^{2}}{(2+3)^{2}}.\nonumber \]

    Solution

    Simplify the numerator and denominator first, then divide.

    \[ \begin{aligned} \frac{(6)^{2}+8^{2}}{(2-7)^{2}} ~ & \textcolor{red}{ \text{ Simplify parentheses in denominator first: } 2 - 7 \text{ simplifies to } -5}\\ \frac{6^{2}+8^{2}}{(-5)^{2}} ~ & \textcolor{red}{ \text{Exponents are next: } (6)^2 \text{ simplifies to } 36,~ 8^2 \text{ simplifies to } 64,~ (- 5)^2 \text{ simplifies to } 25.} \\ \frac{36+64}{25} ~ & \textcolor{red}{ \text{ Add in numerator: } 36 + 64 = 100} \\ \frac{100}{25}  ~ & \textcolor{red}{ \text{ Divide: } 100 \div 25 = 4.}\\ 4  ~ & \textcolor{red}{ \text{ Most Simplified Form }}\end{aligned}\nonumber \]

    Thus, \(\frac{6^{2}+8^{2}}{(2-7)^{2}} \text { simplifies to } 4\).

    Exercise

    Simplify: \(\frac{12+3 \cdot 2}{6}\)Answer

    3

    The Distributive Property

    Consider the expression 2 · (3 + 4). If we follow the “Rules Guiding Order of Operations,” we would evaluate the expression inside the parentheses first. 2 · (3 + 4) = 2 · 7 Parentheses first: 3 + 4 = 7. = 14 Multiply: 2 · 7 = 14.

    However, we could also choose to “distribute” the 2, first multiplying 2 times each addend in the parentheses.

    \[ \begin{aligned} 2 \cdot (3 + 4) = 2 \cdot 3 + 2 \cdot 4 ~ & \textcolor{red}{ \text{ Multiply 2 times both 3 and 4.}} \\ = 6 + 8 ~ & \textcolor{red}{ \text{ Multiply: } 2 \cdot 3 = 6 \text{ and } 2 \cdot 4 = 8.} \\ = 14 ~ & \textcolor{red}{ \text{ Add: } 6 + 8 = 14.} \end{aligned}\nonumber \]

    The fact that we get the same answer in the second approach is an illustration of an important property of whole numbers.1

    The Distributive Property

    Let a, b, and c be any whole numbers. Then,

    a · (b + c) can be distributed to look like this: a · b + a · c.

    We say that “multiplication is distributive with respect to addition.”

    Multiplication is distributive with respect to addition. If you are not computing the product of a number and a sum of numbers, the distributive property does not apply.

    Caution! Wrong Answer Ahead!

    If you are calculating the product of a number and the product of two numbers, the distributive property must not be used. For example, here is a common misapplication of the distributive property.

    \[ \begin{aligned} 2 \cdot (3 \cdot 4) = (2 \cdot 3) \cdot (2 \cdot 4) \\ = 6 \cdot 8 \\ = 48 \end{aligned}\nonumber \]

    This result is quite distant from the correct answer, which is found by computing the product within the parentheses first.

    \[ \begin{aligned} 2 \cdot (3 \cdot 4) = 2 \cdot 12 \\ = 24. \end{aligned}\nonumber \]

    In order to apply the distributive property, you must be multiplying times a sum.

    Example 8

    Use the distributive property to calculate 4 · (5 + 11).

    Solution

    This is the product of a number and a sum, so the distributive property may be applied.

    \[ \begin{aligned} 4 \cdot (5 + 11)~ & \textcolor{red}{ \text{ Distribute the 4 times addend in the sum.}}\\ 4 \cdot 5 + 4 \cdot 11 ~ & \textcolor{red}{ \text{ Multiply: } 4 \cdot 5 \text{ gives } 20 \text{ and } 4 \cdot 11 \text{ gives } 44.}  \\ 20 + 44 ~ & \textcolor{red}{ \text{ Add: } 20 + 44 \text{ gives } 64.} \\ 64 ~ & \textcolor{red}{ \text{ Most Simplified Form}} \end{aligned}\nonumber \]

    Readers should check that the same answer is found by computing the sum within the parentheses first.

    Exercise

    Distribute: 5 · (11 + 8).

    Answer

    95

    The distributive property is the underpinning of the multiplication algorithm learned in our childhood years.

    Example 9

    Multiply: 6 · 43.

    Solution

    We’ll express 43 as sum, then use the distributive property.

    \[ \begin{aligned} 6 \cdot 43 ~ & \textcolor{red}{ \text{ Express 43 as a sum: } 43 = 40 + 3. \text{ Or use standard multiplication as explained below. }}\\ 6 \cdot (40 + 3) ~ & \textcolor{red}{ \text{ Distribute the 6.}} \\ 6 \cdot 40 + 6 \cdot 3 ~ & \textcolor{red}{ \text{ Multiply: } 6 \cdot 40 \text{ to get } 240 \text{ and } 6 \cdot 3 = 18.} \\ 240 + 18 ~ & \textcolor{red}{ \text{ Add: } 240 + 18 = 258.}\\ 258 ~ & \textcolor{red}{ \text{ Most Simplified Form }} \end{aligned}\nonumber \]

    Readers should be able to see this application of the distributive property in the more familiar algorithmic form:

    \( \begin{array}{r}{43} \\ { \times 6} \\ \hline 18 \\ {\frac{240}{258}}\end{array}\)

    Or in the even more condensed form with “carrying:”

    \( \begin{array}{r}{^{1} 43} \\ {\frac{ \times 6}{258}}\end{array}\)

    Exercise

    Use the distributive property to evaluate 8 · 92.

    Answer

    736

    Multiplication is also distributive with respect to subtraction.

    The Distributive Property (Subtraction)

    Let a, b, and c be any whole numbers. Then,

    a · (b − c) distributing will yield a · b − a · c.

    We say the multiplication is “distributive with respect to subtraction.”

    Example 10

    Use the distributive property to simplify: 3 · (12 − 8).

    Solution

    This is the product of a number and a difference, so the distributive property may be applied.

    \[ \begin{aligned} 3 \cdot (12 - 8) ~ & \textcolor{red}{ \text{ Distribute the 3 times each term in the difference.}} \\3 \cdot 12 - 3 \cdot 8 ~ & \textcolor{red}{ \text{Multiply: } 3 \cdot 12 \text{ simplifies to } 36 \text{ and } 3 \cdot 8 \text{ simplifies to } 24.} \\ 36 - 24 ~ & \textcolor{red}{ \text{Subtract: } 36 - 24 \text{ simplifies to } 12.}\\ 12 ~ & \textcolor{red}{ \text{Simplified Form} } \end{aligned}\nonumber \]

    Alternate solution

    Note what happens if we use the usual “order of operations” to evaluate the expression.

    \[ \begin{aligned} 3 \cdot (12 - 8) = 3 \cdot 4 ~ & \textcolor{red}{ \text{ Parentheses first: } 12 - 8 = 4.} \\ = 12 ~ & \textcolor{red}{ \text{ Multiply: } 3 \cdot 4 = 12.} \end{aligned}\nonumber \]

    Same answer.

    Exercise

    Distribute: 8 · (9 − 2).

    Answer

    56

    Exercises

    In Exercises 1-12, simplify the given expression.

    1. 5+2 · 2

    2. 5+2 · 8

    3. 23 − 7 · 2

    4. 37 − 3 · 7

    5. 4 · 3+2 · 5

    6. 2 · 5+9 · 7

    7. 6 · 5+4 · 3

    8. 5 · 2+9 · 8

    9. 9+2 · 3

    10. 3+6 · 6

    11. 32 − 8 · 2

    12. 24 − 2 · 5

    In Exercises 13-28, simplify the given expression.

    13. 45 ÷ 3 · 5

    14. 20 ÷ 1 · 4

    15. 2 · 9 ÷ 3 · 18

    16. 19 · 20 ÷ 4 · 16

    17. 30 ÷ 2 · 3

    18. 27 ÷ 3 · 3

    19. 8 − 6+1

    20. 15 − 5 + 10

    21. 14 · 16 ÷ 16 · 19

    22. 20 · 17 ÷ 17 · 14

    23. 15 · 17 + 10 ÷ 10 − 12 · 4

    24. 14 · 18 + 9 ÷ 3 − 7 · 13

    25. 22 − 10 + 7

    26. 29 − 11 + 1

    27. 20 · 10 + 15 ÷ 5 − 7 · 6

    28. 18 · 19 + 18 ÷ 18 − 6 · 7

    In Exercises 29-40, simplify the given expression.

    29. 9+8 ÷ {4+4}

    30. 10 + 20 ÷ {2+2}

    31. 7 · [8 − 5] − 10

    32. 11 · [12 − 4] − 10

    33. (18 + 10) ÷ (2 + 2)

    34. (14 + 7) ÷ (2 + 5)

    35. 9 · (10 + 7) − 3 · (4 + 10)

    36. 9 · (7 + 7) − 8 · (3 + 8)

    37. 2 · {8 + 12} ÷ 4

    38. 4 · {8+7} ÷ 3

    39. 9+6 · (12 + 3)

    40. 3+5 · (10 + 12)

    In Exercises 41-56, simplify the given expression.

    41. 2+9 · [7 + 3 · (9 + 5)]

    42. 6+3 · [4 + 4 · (5 + 8)]

    43. 7+3 · [8 + 8 · (5 + 9)]

    44. 4+9 · [7 + 6 · (3 + 3)]

    45. 6 − 5[11 − (2 + 8)]

    46. 15 − 1[19 − (7 + 3)]

    47. 11 − 1[19 − (2 + 15)]

    48. 9 − 8[6 − (2 + 3)]

    49. 4{7[9 + 3] − 2[3 + 2]}

    50. 4{8[3 + 9] − 4[6 + 2]}

    51. 9 · [3 + 4 · (5 + 2)]

    52. 3 · [4 + 9 · (8 + 5)]

    53. 3{8[6 + 5] − 8[7 + 3]}

    54. 2{4[6 + 9] − 2[3 + 4]}

    55. 3 · [2 + 4 · (9 + 6)]

    56. 8 · [3 + 9 · (5 + 2)]

    In Exercises 57-68, simplfiy the given expression.

    57. (5 − 2)2

    58. (5 − 3)4

    59. (4 + 2)2

    60. (3 + 5)2

    61. 23 + 33

    62. 54 + 24

    63. 23 − 13

    64. 32 − 12

    65. 12 · 52 + 8 · 9+4

    66. 6 · 32 + 7 · 5 + 12

    67. 9 − 3 · 2 + 12 · 102

    68. 11 − 2 · 3 + 12 · 42

    In Exercises 69-80, simplify the given expression.

    69. 42 − (13 + 2)

    70. 33 − (7 + 6)

    71. 33 − (7 + 12)

    72. 43 − (6 + 5)

    73. 19 + 3[12 − (23 + 1)]

    74. 13 + 12[14 − (22 + 1)]

    75. 17 + 7[13 − (22 + 6)]

    76. 10 + 1[16 − (22 + 9)]

    77. 43 − (12 + 1)

    78. 53 − (17 + 15)

    79. 5 + 7[11 − (22 + 1)]

    80. 10 + 11[20 − (22 + 1)]

    In Exercises 81-92, simplify the given expression.

    81. \( \frac{13+35}{3(4)}\)

    82. \( \frac{35+28}{7(3)}\)

    83. \( \frac{64-(8 \cdot 6-3)}{4 \cdot 7-9}\)

    84. \( \frac{19-(4 \cdot 3-2)}{6 \cdot 3-9}\)

    85. \(\frac{2+13}{4-1}\)

    86. \( \frac{7+1}{8-4}\)

    87. \( \frac{17+14}{9-8}\)

    88. \( \frac{16+2}{13-11}\)

    89. \( \frac{37+27}{8(2)}\)

    90. \( \frac{16+38}{6(3)}\)

    91. \( \frac{40-(3 \cdot 7-9)}{8 \cdot 2-2}\)

    92. \( \frac{60-(8 \cdot 6-3)}{5 \cdot 4-5}\)

    In Exercises 93-100, use the distributive property to evaluate the given expression.

    93. 5 · (8 + 4)

    94. 8 · (4 + 2)

    95. 7 · (8 − 3)

    96. 8 · (9 − 7)

    97. 6 · (7 − 2)

    98. 4 · (8 − 6)

    99. 4 · (3 + 2)

    100. 4 · (9 + 6)

    In Exercises 101-104, use the distributive property to evaluate the given expression using the technique shown in Example 9.

    101. 9 · 62

    102. 3 · 76

    103. 3 · 58

    104. 7 · 57

    Answers

    1. 9

    3. 9

    5. 22

    7. 42

    9. 15

    11. 16

    13. 75

    15. 108

    17. 45

    19. 3

    21. 266

    23. 208

    25. 19

    27. 161

    29. 10

    31. 11

    33. 7

    35. 111

    37. 10

    39. 99

    41. 443

    43. 367

    45. 1

    47. 9

    49. 296

    51. 279

    53. 24

    55. 186

    57. 9

    59. 36

    61. 35

    63. 7

    65. 376

    67. 1203

    69. 1

    71. 8

    73. 28

    75. 38

    77. 51

    79. 47

    81. 4

    83. 1

    85. 5

    87. 31

    89. 4

    91. 2

    93. 60

    95. 35

    97. 30

    99. 20

    101. 558

    103. 174

    1Later, we’ll see that this property applies to all numbers, not just whole numbers

    This page titled 2.4: Order of Operations is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by David Arnold.