CCBC Math 081
Third Edition
Order of Operations
Section 1.7
7 Pages
1.7 Order of Operations
Now you know how to perform all the operations – addition, subtraction, multiplication,
division, exponents, and roots. But what if we have a problem that contains more than one
operation? For instance, what if we have the problem 5  3  4 ? What operation should we
perform first – the addition, the exponent, or the multiplication? It is important that everyone do
the problem the same way in order to get the same answer.
2
For this reason, mathematicians developed a set of rules for evaluating problems that involve
more than one arithmetic operation. The rules, called the Order of Operations, specify the order
in which the computations should be performed. The Order of Operations is given below. It is
important to follow these rules, one step at a time, in the order in which they are presented.
ORDER OF OPERATIONS
Step 1: Parentheses
If there are any operations in parentheses, those computations should be performed first.
Step 2: Exponents and Roots
Simplify any numbers being raised to a power and any numbers under the
symbol.
Step 3: Multiplication and Division
Do these two operations in the order in which they appear, working from left to right.
Step 4: Addition and Subtraction
Do these two operations in the order in which they appear, working from left to right.
To help remember the Order of Operations, try using the phrase in the box on the left below.
Please
Excuse
My Dear
Aunt Sally
Parentheses
Exponents and Roots
Multiplication and Division . . . working from left to right
Addition and Subtraction . . . working from left to right
61
CCBC Math 081
Third Edition
Order of Operations
Section 1.7
7 Pages
Before we present some examples using the Order of Operations, keep in mind that the sign of a
number always precedes the number. Consider the following problems.
5  22
5  (2) 2
5  (2) 2
 54
 54
 54
1
1
9
Evaluate 5  3  4 .
Step 1 – Parentheses. There are none.
5 32 4
2
Example 1:
5  32  4
Step 2 – Exponents and Roots. We have an exponent, 3 gives 9.
 5 9× 4
Step 3 – Multiplication and Division. We multiply 9  4 and get 36.
5+36
Step 4 – Addition and Subtraction. We add 5  36 and get 41.
41
The answer is 41.
2
Practice 1:
Evaluate 23  30  5 .
Watch it:
http://youtu.be/ERXOGK6zBkw
Example 2:
Evaluate 8  5  (4  9) .
Answer:
14
 8  5 (4 9)
Step 1 – Parentheses. We do the multiplication in the parentheses,
4 9 gives 36.
 85  36
Step 2 – Exponents and Roots. There are none.
 85  36
Step 3 – Multiplication and Division. There are none.
- 8+536
Step 4 – Addition and Subtraction. Since the addition comes before
the subtraction, we add 8  5 and get –3.
-3-36
Step 4 – Addition and Subtraction (continued). We change the
subtraction problem to an addition problem by adding the opposite.
-3+-36
Step 4 –Addition and Subtraction (continued). Now we add.
39
The answer is – 39.
62
CCBC Math 081
Third Edition
Order of Operations
Practice 2:
Evaluate 7  4  (8 3) .
Watch it:
http://youtu.be/9wbr3PBo1Ko
Example 3:
Evaluate 80  4(2)  9 .
Section 1.7
7 Pages
Answer:
-21
Remember that the 4(2) means multiplication.
80  4(2)  9
Step 1 – Parentheses. There are no operations in parentheses.
 80  4(2) 
Step 2 – Exponents and Roots. We have a root,
9
9 is 3.
 80÷4(2)3
Step 3 – Multiplication and Division. Working left to right,
the division comes first. We divide 80  4 and get 20.
20(-2) 3
Step 3 – Multiplication and Division (continued). We multiply
20(2) and get –40.
-40+3
Step 4 – Addition and Subtraction. Add (40)  3 and get – 37.
37
The answer is – 37.
Practice 3:
Evaluate 45  5(3)  25 .
Watch it:
http://youtu.be/ebYCY5KYyYE
Example 4:
2
Evaluate  4  (2  14)  ( 3) .
Answer:
-22
 42  (2-14)  (3)
Step 1 – Parentheses. There is a subtraction in the parentheses.
Change the subtraction problem 2  14 to an addition problem.
 42  (2+-14)  (3)
Step 1 – Parentheses (continued). Do the addition problem in
the parentheses. Add 2  (14) to get –12.
-42 (12)  (3)
Step 2 – Exponents and Roots. Evaluate 42 and keep the
negative sign in front. We get –16.
  16 
Step 3 – Multiplication and Division. We divide (12)  (3)
which gives 4.
(-12)÷(-3)
 -16 -4
Step 4 – Addition and Subtraction. We will change the
subtraction problem to an addition problem.
-16+ (-4)
Step 4 – Addition and Subtraction (continued). Now add.
20
The answer is –20.
63
CCBC Math 081
Third Edition
Order of Operations
Practice 4:
2
Evaluate 5  (5  12) (3) .
Watch it:
http://youtu.be/RvZcaTsL5mc
Example 5:
Evaluate 32  (6  4  64)  (5)  2 .
32  (6  4 
64 )  (5)  2
Section 1.7
7 Pages
Answer:
-46
Step 1 – Parentheses. There is a subtraction, an addition,
and a root in the parentheses. Do the root first,
64 is 8.
 32  (6 - 48)  (5)  2
Step 1 – Parentheses (continued). There is a subtraction and
an addition in the parentheses. We do the subtraction since
it comes first. Subtract 6  4 to get 2.
 32 (2+8)(5)  2
Step 1 – Parentheses (continued). There is an addition left in
the parentheses. Add 2  8 which is 10.
32 (10)(5)  2
Step 2 – Exponents and Roots. Evaluate 32 to get 9.
9 (10)×(-5) 2
Step 3 – Multiplication and Division. Do the multiplication
since it comes before the division. So, 10  (5) gives –50.
9(-50) ÷2
Step 3 – Multiplication and Division (continued). Divide
(50)  2 to get –25.
9-(-25)
Step 4 – Addition and Subtraction. We will change the
subtraction problem to an addition problem.
9+25
Step 4 – Addition and Subtraction (continued). Now add.
34
The answer is 34.
16  (8  5  32 )  (3)  2 .
Practice 5:
Evaluate
Watch it:
http://youtu.be/S5tPbqzNt5E
64
Answer:
12
CCBC Math 081
Third Edition
Example 6:
Order of Operations
Section 1.7
7 Pages
Evaluate 3  4  5 .
This problem contains the absolute value symbol. Where does this fit in the order of operations?
If you look back, you will find that it is not specifically mentioned in the Order of Operations
steps.
However, in Section 1.2 of this chapter we learned that the absolute value symbol is treated as
parentheses for the purpose of order of operations. This is because it functions as a grouping
symbol just as parentheses do. Therefore, absolute value is included in Step 1 of the Order of
Operations.

-3 - 4  5

-3+ -4  5
Step 1 – Parentheses (continued). Do the addition problem inside the
absolute value symbol. Add 3  4 to get –7.

 -7  
Step 1 – Parentheses (continued). Since there is just one number inside the
absolute value symbol, we now apply the definition of absolute value. We
know that the absolute value of any non-zero number is positive. So, 7  7
  
Step 1 – Parentheses. There are no parentheses, but there is an absolute
value symbol. It is treated as parentheses for the purpose of order of
operations. The absolute value acts as a grouping symbol, so the operation
within the absolute value symbol must be performed before we apply the
definition of absolute value. In this case, we must do the subtraction problem
3  4 first. How? By changing it to an addition problem.
Step 2 – Exponents and Roots. There are no exponents or roots.
Step 3 – Multiplication and Division. There is nothing to multiply or divide.

7+5
=
12
Step 4 – Addition and Subtraction. We do the addition problem and get 12.
The answer is 12.
Practice 6:
Evaluate 5 | 3  6 | .
Watch it:
http://youtu.be/8kOVJX8lD_s
Answer:
8
In all these example problems, we performed one operation at a time, and we were careful to
bring down the rest of the problem to the next step. It is very important for you to do the same as
you complete the practice exercises that follow.
Watch All: http://youtu.be/h0XUAeHKJRs
65
CCBC Math 081
Third Edition
Order of Operations
Section 1.7
7 Pages
1.7 Order of Operations Exercises
Evaluate each of the following expressions. Remember that multiplication may be represented
with parentheses, “ ” or “  ” symbols.
1.
3  5  8
15.
16  2  9  ( 3)
2.
15  ( 3)  10
16.
4  2  7  32
3.
5  112  3
17.
4  (2  7)  25
4.
62  4  2
18.
3  23  5  40
5.
3  (6)2  20
19.
 4(32 )  15  ( 4)
6.
52  36
20.
22  5  (3  1)
7.
8  (15  3)  4
21.
30  6  9
8.
2  (8  10)  2
22.
15  (23 )  (4)(3)
9.
6  2  (1  5)
23.
20  2  22  5  (2)
10.
6(3  8  4)
24.
42  (8) 
11.
4  5  (6  2)
25.
8  24  (25  5)  81
26.
 3 (2  9)  4 (7  2)
27.
4  6  (2  7)2  (17 12)
28.
6
 8   9(8)
12.
6
13.
9  4(2  3)
14.
2  8  4

16  52

3
66

3

27  2  4  (16  9)
CCBC Math 081
Third Edition
Order of Operations
Section 1.7
7 Pages
1.7 Order of Operation Exercises Answers
1.
37
15.
11
2.
5
16.
– 19
3.
123
17.
–4
4.
38
18.
29
5.
32
19.
– 25
6.
31
20.
– 14
7.
24
21.
10
8.
3
22.
21
9.
14
23.
0
10.
30
24.
– 23
11.
– 24
25.
124
12.
84
26.
– 15
13.
13
27.
– 26
14.
40
28.
– 109
67
CCBC Math 081
Third Edition
Chapter 1 Summary
CHAPTER 1 SUMMARY
Integers
Section
1.1
Natural Numbers
{1, 2, 3, 4, 5 … }
Integers
{…, -4, -3, -2, -1, 0, 1, 2, 3, 4, … }
Whole Numbers
{0, 1, 2, 3, 4, 5 … }
Negative
Section
The Absolute Value of 0 is 0.
Section
1.3
The Absolute Value of all other real numbers is positive.
0 0
1.2
Positive
6 6
6  6
Addition
SSS Method
5  3
53  8
5  3  8
 Same – signs of #s are the same
 Sum – add absolute values
 Same – answer has same sign as original #s
DDD Method
4  (6)
64  2
4  (6)  2
 Different – signs of #s are different
 Difference – subtract absolute values
 Dominant – answer has sign of bigger #
Section
1.4
Subtraction – use the “Add the Opposite” method
Subtracting a Positive becomes Adding a Negative:
Subtracting a Negative becomes Adding a Positive:
5  8  5  (8)
5  (8)  5  8
Multiplication and Division
Section
1.5
a  b  a  ( b)
a  (b)  a  b
Same Signs
Positive answer
Different Signs
Negative answer
(3) (4)  12
(12)  (3)  4
3 (4)  12
12  3  4
68
Use of Zero
n 00
0n  0
n  0  undefined
CCBC Math 081
Third Edition
Section
1.6
Chapter 1 Summary
Exponents a n - the exponent (n) tells how many times the base (a) is multiplied with itself
Roots
3
Section
1.7
(6)2  (6) (6)  36
62  6 6  36
n
62  (6 6)  36
Note: a 0  1
a = the number that, when raised to the power n, gives a
8 means
 3  . The answer is 2 because 23  2 2 2  8 .
Order of Operations
Please
Excuse
My Dear
Aunt Sally
Parentheses
Exponents and Roots
Multiplication and Division (working left to right)
Addition and Subtraction (working left to right)
69
4  (9  7)3  5
 4  23  5
 4  8 5
 4  40
 44
CCBC Math 081
Third Edition
Chapter 1 Review
CHAPTER 1
Chapter Review
1. Circle each number system that the number –34 belongs to:
Natural Numbers
Whole Numbers
Integers
2. Circle all the positive integers in the list of numbers:
1.2
0
 5.8
7
A
3. What is the value of each of the points
named by the capital letters on the
number line?
4. Place the numbers 4 and –1 on the number line:
0
5.
 8 
18.
53 
6.
27  (14) 
19.
170 
7.
38  56 
20.
122 
8.
24  (24) 
21.
(12)2 
9.
20  8 
22.
(5)3 
17  (9) 
23.
3
10.
11.
6 12 
12.
42  (7) 
13.
90  (3) 
14.
14  0 
15.
0  56 
16.
12  (4) 
17.
5(8)(4) 
70
27 
49 
25.
61  3  4 
26.
5
27.
52  3(1  4)  40 
28.
22  3 8  23  
29.
50  2  52  9 
30.
4  6  72 

1
2
3
B
0
24.
9

81  4  5  7 
6
CCBC Math 081
Third Edition
Chapter 1 Review
Chapter 1 Review Answers
1.
Natural Numbers
2.
1.2
3.
A = –3
4.
0
7
Whole Numbers
 5.8
9
1
2
3
Integers
6
B= 5
-1 0
4
5.
–8
18.
125
6.
–41
19.
1
7.
18
20.
–144
8.
0
21.
144
9.
–28
22.
–125
10.
–8
23.
3
11.
–72
24.
7
12.
6
25.
–6
13.
–30
26.
–2
14.
0
27.
24
15.
0
28.
7
16.
48
29.
–50
17.
–160
30.
74
71