4

Order of Operations

In this chapter, you apply your skills in computation to perform a series of indicated numerical operations. This chapter lays the foundation for numerical calculations by introducing you to the order of operations.

Grouping Symbols

Grouping symbols such as parentheses ( ), brackets [ ], and braces { } are used to keep things together that belong together. Fraction bars and absolute value bars are grouping symbols as well. When you do computations in numerical expressions, do operations in grouping symbols first. It is very important that you do so when you have addition or subtraction inside the grouping symbol.

Keep in mind, however, that parentheses also are used to indicate multiplication as in (−5)(−8) or for clarity as in −(−35).

Fraction bars indicate division. For 18 instance, means 18 ÷ −2

Grouping symbols say “Do me first!”

Problem

Evaluate each expression.

a. (1 + 1)⁴

b.

c.

d. |8 + −15|

Solution

a. (1 + 1)⁴

Step 1.   Parentheses are a grouping symbol, so do 1 + 1 first.

(1 + 1)⁴ = 2⁴

Step 2.   Evaluate 2⁴.

2⁴ = 16

Step 3.   Review the main results.

(1 + 1)⁴ = 2⁴ = 16

When you no longer need the grouping symbol, omit it.

(1 + 1)⁴ ≠ 1⁴ + 1⁴; (1 + 1)⁴ = 16, but 1⁴ + 1⁴ = 1 + 1 = 2. Not performing the addition, 1 + 1, inside the parentheses first can lead to an incorrect result.

b.

Step 1.   The fraction bar is a grouping symbol, so do the addition, 4 + 10, over the fraction bar first.

Step 2.   Simplify .

. Not performing the addition, 4 + 10, first can lead to an incorrect result.

Step 3.   Review the main results.

c.

Step 1.   The fraction bar is a grouping symbol, so do the addition, −7 + 25, over the fraction bar and the subtraction, 3 - 5, under the fraction bar first.

Step 2.   Compute .

Step 3.   Review the main results.

, but . Not performing the addition, −7 + 25, and the subtraction, 3 − 5, first can result in a cancellation error.

d. |8 + −15|

Step 1.   Absolute value bars are a grouping symbol, so do 8 + −15 first.

|8 + −15| =|−7|

Step 2.   Evaluate | − 7|.

= 7

Step 3.   Review the main results.

|8 + −15 =|−7| = 7

8 + −15|≠ |8 + |−15|; |8 + −15| = 7, but 8 + |−15| = 8 + 15 = 23. Not performing the addition, 8 + −15, first can lead to an incorrect result.

PEMDAS

Mathematical expressions like 60 ÷ 12 − 3 × 4 + (1 + 1)³ and are numerical expressions. When you evaluate numerical expressions by doing the indicated math, you must follow the order of operations. Use the mnemonic “Please Excuse My Dear Aunt Sally”—abbreviated as PE(MD)(AS)—to help you remember the following order.

Order of Operations

1. Do computations inside Parentheses (or other grouping symbols).

2. Do Exponents (also, evaluate absolute values and roots).

3. Do Multiplication and Division, in the order in which they occur from left to right.

4. Do Addition and Subtraction, in the order in which they occur from left to right.

In the order of operations, multiplication does not always have to be done before division, or addition before subtraction. You multiply and divide in the order in which they occur in the problem. Similarly, you add and subtract in the order in which they occur in the problem.

Problem

Evaluate the expression.

a. 60 + −12 - 3 × 4 + (1 + 1)³

b. 100 + 8 × 3² − 63 + (2 + 5)

c.

d. (12−5)−(5−12)

e. −8 + 2 (−1)² + 6

f. 2 + 10 × 3 +1

g. (2 × 5)² + 2 × 5²

h. 2 + 5(6 − 4)

Solution

a. 60 ÷ 12 − 3 × 4 + (1 + 1)³

Step 1.   Do 1 + 1 inside the parentheses.

= 60 ÷ 12 − 3 × 4 + 2³

Step 2.   Do the exponent: 2³.

= 60 ÷ 12 − 3 × 4 + 8

Step 3.   Do multiplication and division, from left to right.

Do 60 ÷ 12 because division occurs first.

= 5 − 3 × 4 + 8

Do 3 × 4.

= 5 − 12 + 8

Step 4.   Do addition and subtraction, from left to right.

Do 5 − 12 because subtraction occurs first.

= −7 + 8

Do −7 + 8.

= 1

5 − 3 × 4 + 8 ≠ 2 × 12. Multiply before adding or subtracting—when no grouping symbols are present.

Step 5.   Review the main results.

60 + 12 − 3 × 4 + (1 + 1)³

= 60 + 12 − 3 × 4 + 2³

= 60 + 12 − 3 × 4 + 8

= 5 − 12 + 8

= 1

b. 100 + 8 × 3² − 63 + (2 + 5)

Step 1.   Compute 2 + 5 inside the parentheses.

= 100 + 8 × 3² − 63 + 7

Step 2.   Do the exponent: 3².

= 100 + 8 × 9 − 63 + 7

Step 3.   Do multiplication and division, from left to ^right.

Do 8 × 9 because multiplication occurs first.

= 100 + 72 − 63 ÷ 7

Do 63 + 7.

= 100 + 72 − 9

Step 4.   Do addition and subtraction, from left to right.

Do 100 + 72 because addition occurs first.

= 172 − 9

Do 172 − 9.

= 163

Step 5.   Review the main results.

100 + 8 × 3² − 63 + (2 + 5)

= 100 + 8 × 3² − 63 + 7

= 100 + 8 × 9 − 63 + 7

= 100 + 72 − 9

= 163

8 × 3² ≠ 24²; 8 × 3² = 8 × 9 = 72, but 24² = 576. Do exponents before multiplication.

100 + 8 × 9 ≠ 108 × 9. Do multiplication before addition (except when a grouping symbol indicates otherwise).

72 − 63 + 7 ≠ 9 ÷ 7. Do division before subtraction (except when a grouping symbol indicates otherwise).

c.

Step 1.   Compute quantities in grouping symbols.

Step 2.   Evaluate |−7| and 2³.

Step 3.   Do division: .

= −9 + 7 − 8

Step 4.   Do addition and subtraction, from left to right.

Do −9 + 7 because addition occurs first.

= −2 − 8

Do −2 − 8.

= −10

Step 5.   Review the main results.

Evaluate absolute value expressions and exponents before multiplication or division.

d. (12 - 5) - (5 −12)

Step 1.   Compute quantities in parentheses.

= 7 −(−7)

Step 2.   Do subtraction: 7 −(−7).

= 7 + 7 = 14

Step 3.   Review the main results.

(12 − 5) − (5 − 12)

= 7 −(−7)

= 7 + 7

= 14

e. −8 + 2(−1)² + 6

Step 1.   Do the exponent: (−1)².

= −8 + 2 · 1 + 6

Step 2.   Do multiplication: 2 · 1.

= −8 + 2 + 6

Step 3.   Do addition.

= −8 + 2 + 6 = −6 + 6 = 0

Step 4.   Review the main results.

−8 + 2(−l)² + 6

= −8 + 2 1 + 6

= −8 + 2 + 6

= 0

2 −1 = 2 × 1. You can use a raised dot to show multiplication.

f. 2 + 10 × 3 + 1

Step 1.   Do multiplication: 10 × 3.

= 2 + 30 + 1

Step 2.   Do addition.

= 2 + 30 + 1 = 32 + 1 = 33

Step 3.   Review the main results.

2 + 10 × 3 + 1

= 2 + 30 + 1

= 33

2 + 10 × 3 + 1 ≠ 12 × 4; 2 + 10 × 3 + 1 = 33, but 12 × 4 = 48. Don’t add (or subtract) before multiplying, unless you have a grouping symbol that tells you to do so.

g. (2 × 5)² + 2 × 5²

Step 1.   Do 2 × 5 inside the parentheses.

= 10² + 2 × 5²

Step 2.   Do the exponents.

= 100 + 2 × 25

Step 3.   Do multiplication: 2 × 25.

= 100 + 50

Step 4.   Do addition.

= 150

Step 5.   Review the main results.

(2 × 5)² + 2 × 5²

= 10² + 2 × 5²

= 100 + 2 × 25 = 100+50

= 150

2 × 5² ≠ 10²; 2 × 5² = 2 × 25 = 50, but 10² = 100.

h. 2 + 5(6 − 4)

Step 1.   Do parentheses.

= 2 + 5 (2)

Step 2.   Do multiplication.

= 2 + 10

Step 3.   Do addition.

= 12

Step 4.   Review the main results.

2 + 5(6 − 4)

= 2 + 5(2)

= 2 + 10

= 12

2 + 5(6 − 4) ≠ 7(6 − 4) because parentheses and multiplication come before addition.

5(2) = 5 × 2. You can use parentheses to show multiplication.

Exercise 4

Evaluate the expression.

1. (5 + 7)6 − 10

2. (−7²)(6 − 8)

3. (2 − 3)(−20)

4.

5.

6. −2² −3 −(15−4)²

7. 5(11 − 3 − 6 · 2)²

8.

9.

10.

11.