Order of Operations - Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST - Easy Algebra Step-by-Step

Easy Algebra Step-by-Step: Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST! (2012)

Chapter 5. 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.


Do keep in mind that parentheses are also used to indicate multiplication, as in (–5)(–8) or for clarity, as in –(–35).


Fraction bars, absolute value bars | |, and square root symbols Image are also grouping symbols. When you are performing computations, perform operations in grouping symbols first.


Grouping symbols say “Do me first!”


It is very important that you do so when you have addition or subtraction inside the grouping symbol.

Problem Simplify.

a. (1 + 1)4

b. Image

c. Image

d. |8 + –5|

e. Image

Solution

a. (1 + 1)4


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


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

(1 + 1)4 = 24

Step 2. Evaluate 24.

= 16

b. Image


(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.


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

Image

Step 2. Simplify Image.

Image

c. Image


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


Image 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.

Image

Step 2. Compute Image.

= –9


Image
Not performing the addition, –7 + 25, and the subtraction, 3 – 5, first can lead to an incorrect result.


d. |8 + –15|

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

|8 + –15| = |–7|

Step 2. Evaluate |–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.


e. Image

Image Step 1. The square root symbol is a grouping symbol, so do 36 + 64 first.

Image

Step 2. Evaluate Image.

= 10


Image, Image. Not performing the addition, 36 + 64, first can lead to an incorrect result.


PEMDAS

You must follow the order of operations to simplify mathematical expressions. Use the mnemonic “Please Excuse My Dear Aunt Sally”—abbreviated as PE(MD)(AS) to help you remember the following order.

Image Order of Operations

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

2. Evaluate Exponential expressions (also, evaluate absolute value, square root, and other root expressions).

3. Perform Multiplication and Division, in the order in which these operations occur from left to right.

4. Perform Addition and Subtraction, in the order in which these operations 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 they occur in the problem. Similarly, you add and subtract in the order they occur in the problem.


Problem Simplify.

a. Image

b. 100 + 8 · 32 – 63 ÷ (2 + 5)

c. Image

Solution

a. Image

Image Step 1. Compute 1 + 1 inside the parentheses.

Image

Step 2. Evaluate 23.

Image

Step 3. Compute Image.

= 5 – 3 · 4 + 8


5 – 3 · 4 + 8 ≠ 2 · 12. Multiply before adding or subtractin—when no grouping symbols are present.


Step 4. Compute 3 · 4.

= 5 – 12 + 8

Step 5. Compute 5 – 12.

= –7 + 8

Step 6. Compute –7 + 8.

= 1

Step 7. Review the main steps.

Image

b. 100 + 8 · 32 – 63 ÷ (2 + 5)


8 · 32 ≠ 242. 8 · 32 = 8 · 9 = 72, but 242 = 576. Do exponentiation before multiplication.


Image Step 1. Compute 2 + 5 inside the parentheses.

100 + 8 · 32 – 63 ÷ (2 + 5)

= 100 + 8 · 32 – 63 ÷ 7


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


Step 2. Evaluate 32.

= 100 + 8 · 9 – 63 ÷ 7

Step 3. Compute 8 · 9.

= 100 + 72 – 63 ÷ 7

Step 4. Compute 63 ÷ 7.

= 100 + 72 – 9


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


Step 5. Compute 100 + 72.

= 172 – 9

Step 6. Compute 172 – 9.

= 163

Step 7. Review the main steps.

100 + 8 · 32 – 63 ÷ (2 + 5) = 100 + 8 · 32 – 63 ÷ 7 = 100 + 8 · 9 – 63 ÷ 7

= 100 + 72 – 9 = 163

c. Image

Image Step 1. Compute quantities in grouping symbols.

Image

Step 2. Evaluate |–7| and 23.

Image


Evaluate absolute value expressions before multiplication or division.


Step 3. Compute Image.

= –9 + 7 – 8

Step 4. Compute –9 + 7.

= –2 – 8

Step 5. Compute –2 – 8.

= –10

Step 6. Review the main steps.

Image

Image Exercise 5

Simplify.

1. (5 + 7)6 – 10

2. (–72)(6 – 8)

3. (2 – 3)(–20)

4. Image

5. Image

6. –22 · –3 – (15 – 4)2

7. 5(11 – 3 – 6 · 2)2

8. Image

9. Image

10. Image

11. Image

12. (12 – 5) – (5 – 12)

13. Image

14. –8 + 2(–1)2 + 6

15. Image