Section 6.3
PRE-ACTIVITY
PREPARATION
Evaluating Expressions
Using the Order of Operations
Sales of admission tickets to the family concert were as follows: 50 adult tickets sold for $5 each, 300 youth
tickets sold for $3 each, and 50 senior citizen tickets sold for $4 each. What was the total of ticket sales in
dollars?
To answer the question, you would simply calculate the revenue from each
type of ticket sold (by multiplying) and then add the three results: 50 × $5
or $250 from the sale of adult tickets, 300 × $3 or $900 from youth tickets,
and 50 × $4 or $200 from senior citizen sales add up to $1350. In other
words, you would follow a sequence of operational steps to arrive at your
answer.
As you progress in your knowledge of mathematics, you will find that the processes you encounter most
frequently will be the basic operations of addition, subtraction, multiplication, division, and the evaluation of
exponential expressions. Your success in mathematics builds on your mastery of these basic skills as well as
your ability to apply them in the correct order when several appear in the same mathematical expression.
LEARNING OBJECTIVE
• Evaluate expressions using the correct order of operations.
TERMINOLOGY
PREVIOUSLY USED
NEW TERMS
evaluate
operation
exponent
simplify
expression
term
TO
LEARN
order of operations
BUILDING MATHEMATICAL LANGUAGE
In mathematical language, you can answer the question in the introduction about ticket sales by setting
up and evaluating an expression:
50 × $5 + 300 × $3 + 50 × $4
= $250 + $900 + $200
= $1350
583
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
584
That is, you follow a sequence of operational steps, multiplying before adding the three components of ticket
sales.
In fact, there is a logical and universally agreed upon mathematical sequence, an order of operations,
used to evaluate any mathematical expression requiring a series of steps. Using this order of operations ensures
the same answer for everyone who evaluates a given expression. It is as follows:
Order of Operations
To evaluate an expression, the simplification process must follow the Order of Operations:
First, simplify the operations within Parentheses.
Then, simplify all numbers with Exponents.
Then, compute Multiplication and Division, left to right as they occur in the expression.
Finally, compute Addition and Subtraction, left to right as they occur in the expression.
METHODOLOGY
Applying the Order of Operations to Simplify an Expression
►
Example 1: Simplify: 75 ÷ 5 × 2 + 23 − 4 (11 − 8)2
►
Example 2: Simplify: (7 + 2)2 + 5 × 9 − 42 + 12 ÷ 4
Steps in the Methodology
Step 1
Identify the
terms.
Use brackets, [ ] or { },
to identify the terms of
the expression. Recall that
addition and subtraction
signs separate the terms
of an expression.
Try It!
Example 1
[75÷5×2]+[2 3]–[4(11–8)2]
The expression is written as a fraction with an
Special
expression in either or both the numerator and
Case:
denominator (see page 588, Model 4)
Example 2
Section 6.3 — Evaluating Expressions Using the Order of Operations
Steps in the Methodology
Step 2
Simplify
operations in
parentheses.
Simplify the operation(s)
within Parentheses, if
there are any, for each
term.
585
Example 1
Example 2
[75÷5×2]+[2 3]–[4(11–8) 2]
=[75÷5×2]+[2 3]–[4(3) 2] P
To assure that you
are doing the steps in
the correct order of
operations, it may be
helpful to label each step
as you compute it.
As each term is simplified
to one number, you
may drop the brackets
surrounding it.
Step 3
Simplify
numbers
with
exponents.
Step 4
Mulitply and
Divide left to
right.
Simplify the numbers
with Exponents, if there
are any, in each term.
=[75÷5×2]+[23]–[4(3)2]
2×2×2
3×3
=[75÷5×2]+ 8 –[4×9] E
Compute Multiplication
and Division, left to right
as they are situated in
each term.
=[75÷5×2]+ 8 –[4×9]
=[15×2]+ 8 –[4×9]
=[30]+ 8 –[36] M & D
Step 5
Add and
Subtract left
to right.
Compute Addition and
Subtraction of the
simplified terms, left to
right as they are situated
in the expression.
Step 6
Present your final answer.
Present the
answer.
Note: If the answer is in
fraction form, reduce it.
= 30+ 8 – 36
= 38 – 36
= 2
A & S
2
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
586
Suggested Validation when
Applying the Order of Operations
When simplifying an expression with the Order of Operations, you can be fully
confident in your answer only when you apply the correct order as well as do
each computation accurately.
Keeping track of your steps by labeling each one as you go is an effective way
to assure that the order is correct. Validating the accuracy of each computation
as you work through the problem can further assure the accuracy of your final
answer.
The following is a sample validation for Example 1 of the Methodology.
75 ÷ 5 × 2 + 23 − 4(11 − 8)2
Validation
[75 ÷ 5 × 2] + [23] − [4(11 − 8)2]
Step 1
of Step 2
Step 2
= [75 ÷ 5 × 2] + [23] −
[4(3)2] P
Step 3
= [75 ÷ 5 × 2] +
[4(9)]
8
−
3 + 8 = 11 9
E
15×2
of Step 4
Step 4
=
30
+
8
−
36
M&D
30 ÷ 2 = 15
15 × 5 = 75 9
36 ÷ 9 = 4 9
of Step 5
Steps 5 & 6
=
2 Answer
Order used: P, E, M & D, A & S
A&S
2 + 36 – 8
= 38 – 8 9
= 30 9
Section 6.3 — Evaluating Expressions Using the Order of Operations
587
MODELS
Model 1
Simplify
Validation:
4.6 ÷ 2 – (0.5)2 + 2 (8 – 1.5)
Step 1: Identify the terms:
[4.6 ÷ 2] – [(0.5)2] + [2 (8 – 1.5)]
of Step 2
Step 2: P
= [4.6 ÷ 2] – [(0.5)2] + [2 (6.5)]
6.5 + 1.5 = 8 9
0.5 × 0.5
Step 3: E
= [4.6 ÷ 2] – [0.25]
+ [2 (6.5)]
of Step 4
Step 4: M & D
=
2.3 – 0.25
+ 13
Step 5: A & S, left to right as they occur
= 2.05 + 13
= 15.05
Step 6:
2.3 × 2 = 4.6 9
13.0 ÷ 6.5 = 29
of Step 5
15.05 – 13 + 0.25
= 2.05 + 0.25 = 2.30 9
Order: P, E, M&D, A&S 9
Answer: 15.05
Model 2
Simplify
Validation:
5 (–4) – (6 – 1)3 – 7 (–2)
Step 1: Identify the terms:
[5(–4)] – [(6 – 1)3] – [7 (–2)]
of Step 2
Step 2: P
= [5(–4)] –
[(5)3]
– [7 (–2)]
5+1=6 9
5×5×5
Step 3: E
= [5(–4)] –
125
– [7 (–2)]
of Step 4
Step 4: M
=
–20 –
125
– (–14)
Step 5: S, change all subtraction to addition
= –20 + (–125) + (+14)
= –145 + (+14)
= –131
Step 6:
Answer: –131
–20 ÷ (–4) = +5 9
–14 ÷ (–2) = +7 9
of Step 5
–131 + (–14) + 125
= –145 + 125 = –20 9
Order: P, E, M, S 9
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
588
Model 3
⎛1⎞
2 3
Simplify: ⎜⎜⎜ ⎟⎟⎟ + − × 10
⎝ 2⎠
3 5
2
Validation:
⎡⎛ 1 ⎞2 ⎤
⎢⎜ ⎟⎟ ⎥ + ⎡⎢ 2 ⎤⎥ − ⎡⎢ 3 × 10⎤⎥
Step 1:
⎢⎜⎜⎝ 2 ⎟⎠ ⎥
⎥⎦
⎢⎣ 5
⎢⎣ 3 ⎥⎦
⎢⎣
⎥⎦
Step 2: skip this step—no operations inside Parentheses
1 1
×
2 2
Step 3: E
Step 4: M
=
=
1
4
1
4
⎤
⎡3
⎡2⎤
+ ⎢ ⎥ − ⎢ × 10⎥
⎥⎦
⎢⎣ 5
⎢⎣ 3 ⎥⎦
⎡2⎤
+ ⎢ ⎥ −
⎢⎣ 3 ⎥⎦
3
1
5
2
×
10 6
= =6
1
1
=
6
3
=
9
10 5
of Step 5
11
1
− 6 = −5
12
12
Answer : – 5
Step 6:
6 ÷ 10 =
6
Step 5: A & S, left to right as they occur
1 2
3
8
11
+ =
+ =
4 3 12 12 12
of Step 4
–5
11 72 11 ⎛⎜ 72 ⎞⎟
− =
+ ⎜− ⎟
12 12 12 ⎜⎝ 12 ⎟⎠
11 + (−72)
61
1
=
= − = −5
12
12
12
1
12
1
11
+6 = +
12
12
11 2 11
8
− =
−
12 3 12 12
3
1
9
=
=
12
4
Order: E, M, A & S 9
Special Case: The Expression is Written as a Fraction with an Expression
in Either or Both the Numerator and Denominator
Model 4
A
►
Simplify:
4 (2 − 7) − 2 (5 − 2)
−10 − 2 − 1
The fraction bar indicates that the numerator and denominator are to be treated as two separate
expressions. Simplify each expression separately, following the Order of Operations procedure;
then reduce the resulting fraction, paying careful attention to the correct sign of the answer.
⎡4 (2 –7)⎤ − ⎡2 (5 –2)⎤
⎣
⎦
⎣
⎦
−10 − 2 − 1
Step 1: Identify the terms:
Step 2: P
⎡4 (– 5)⎤ − ⎡2 (3)⎤
⎦
⎣
⎦
= ⎣
−10 − 2 − 1
Step 4: M
Step 5: S change to addition =
Step 6: Reduce: −26 = +2
−13
of Step 2
–5 + 7 = 2 9
3+2=59
of Step 4
Step 3: no Exponents, skip this step
=
Validation:
−20 − 6
−10 − 2 − 1
−20 + (−6)
−10 +
(−2)
Answer: 2
−26
=
+ (−1) −13
–20 ÷ (–5) = +4 9
6 ÷ 3 = 29
of Step 5
–26 + 6 = –20 9
–13 + 1 + 2
= –12 + 2 = –10 9
Order: P, M, S 9
Section 6.3 — Evaluating Expressions Using the Order of Operations
►
3 (5) + 32 + 3 (−7)
Simplify:
B
589
Validation:
−14 − 2 (−1)
⎡3 (5)⎤ +
⎣
⎦
[−14]
Step 1: Identify the terms:
⎡32 ⎤
⎢⎣ ⎥⎦
−
+ ⎡⎣3 (−7)⎤⎦
⎡2 (−1)⎤
⎣
⎦
Step 2: skip this step—no operations inside Parentheses
3× 3
Step 3: E
=
Step 4: M
=
=
Step 5: A & S
Step 6: Reduce:
⎡3 (5)⎤ + 9 + ⎡3 (−7)⎤
⎣
⎦
⎣
⎦
⎡
⎤
14
2
1
−
−
−
[
]
⎣ ( )⎦
15 + 9 +
−14 −
(−21)
(−2)
+3
3
=
−14 + (+2) −12
3
3
1
=−
=−
−12
12
4
Answer : –
1
4
of Step 4
15 ÷ 5 = 3 9
–21 ÷ (–7) = +3 9
–2 ÷ (–1) = +2 9
of Step 5
3 – (–21) – 9
= 3 + (+21) + (–9)
=24 + (–9) = 15 9
–12 + (–2) = –14 9
Order: E, M, A & S 9
ADDRESSING COMMON ERRORS
Issue
Incorrectly
identifying the
terms of an
expression
Incorrect
Process
Simplify:
52 − 2 (−3 + 9)
= 5 − 2 ((6)
2
=2
25−
− 2 (6)
6
= 23
3 (6
(6)
6)
= 138
Resolution
Insert brackets to
separate the terms.
The terms of an
expression are
separated by addition
and subtraction signs.
Simplify each term
separately.
Correct Process
Validation
Simplify:
[52] −
– [2 (−3 + 9)]
= [52] −
– [2 (6)] P
6 − 9 = −3 9
5×5
= 25 −
– [2 (6)]
E
= 25 −
– 12
M
12 ÷ 6 = 2 9
= 13
S
13 +12 = 25 9
PEMS,
correct order
Not multiplying
and dividing
in the correct
order when
simplifying a
term in which
they both
occur
Simplify:
3(−24)
−24
2 ÷ 6 (4)
24)
= –7
–72 ÷ 24
2
= –3
When multiplication and
division occur within the
same term, follow the
order of operations for
that term, multiplying
and dividing left to
right.
Simplify:
3(−24) ÷ 6 (4)
–48÷4×6÷(–24)
This is a single term
(no addition or
subtraction signs).
= –12×6÷(–24)
Left to right:
= –72 ÷ 6 (4)
= –12 (4)
= –48
= –72÷(–24)
=39
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
590
Issue
Evaluate:
1 +8
6 − 13
= 6 − 13 + 8
}
Not adding and
subtracting
from left to
right in the
final step
Incorrect
Process
= 6 − 21
= −15
Resolution
Once the terms have all
been simplified to one
number each, addition
and subtraction must
be done left to right as
they occur to comply
with the Order of
Operations.
Correct Process
Validation
Evaluate:
6 −
– 13 +
+ 8
Left to right:
= 6 + (–13) + 8
=
–7 + 8
=
+1
1 − 8 + 13
= 1 + (−8) + 13
= −7 + 13
=69
At this point, unless
you convert each
subtraction to addition
of the opposite,
you cannot use the
Associative Property for
Addition.
That is, for example,
6–13+8 ‡ 6–(13+8)
Making
arithmetic
and/or sign
errors
Simplify:
–2(–8
–8
8 + 3) + 6
= –2
2 (5)) + 6
= –10 + 6
= –4
Making arithmetic and/
or sign errors in any
step will, of course,
ultimately result in an
incorrect answer.
Validate each step as
you work through the
problem, so as not to
carry through an earlier
error.
Simplify:
–2 (–8 + 3) +
+ 6
= [–2 (–5)] + 6 P
–5 – 3
= –5 + (–3)
= –8 9
= +10 + 6
M
10 ÷ (–5) = –29
= 16
A
16 – 6 = 10 9
[–2 (–8 + 3)] + 6
Order: P, M, A 9
PREPARATION INVENTORY
Before proceeding, you should have an understanding of each of the following:
the terminology and notation associated with order of operations problems
the purpose for an agreed upon Order of Operations
how to apply the Order of Operations to simplify an expression with multiple operations
validation techniques for Order of Operations problems
Section 6.3
ACTIVITY
Evaluating Expressions
Using the Order of Operations
PERFORMANCE CRITERIA
• Evaluating expressions
– accuracy
– documentation of steps
CRITICAL THINKING QUESTIONS
1. What is the order of operations to follow when evaluating an expression?
2. What is the purpose of the Order of Operations?
3. How can you identify the terms of an expression? Do all expressions have terms?
4. When computing a series of multiplication and division operations within a single term, in what order must
they be done?
591
592
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
5. In computing a series of addition and subtraction operations, in what order must they be done and why?
6. What is a strategy you can use to validate an order of operations problem?
7. Why do you think the operations in parentheses are done before exponents and the exponents before
multiplication and division?
8. Why do you think multiplication and division are done before addition and subtraction when simplifying an
expression?
Section 6.3 — Evaluating Expressions Using the Order of Operations
TIPS
FOR
593
SUCCESS
• Separate terms with brackets and/or highlight the addition and subtraction signs separating the terms.
• Once identified, terms can be simplified simultaneously (for example, if several terms have an exponent to
evaluate).
• To assure accuracy, bring the entire expression down to the next line with each simplification.
• Label which step you are doing to document and assure the correct order. (P, E, M or D, A or S)
• Validate each step for computational accuracy.
• When you simplify an expression that begins as a fraction of two expressions,
→if the answer can be converted to an integer, it must be presented as an integer. For example, 4 = −2
−2
→if the answer simplifies to a negative number over a negative number, the final answer is positive.
For example, −2 = + 2 or 2 (A negative number divided by a negative number is always positive.)
−3
3
3
DEMONSTRATE YOUR UNDERSTANDING
Simplify each of the following expressions:
Expression
1)
16 – 23 ÷ 4 (2)
2)
32 × 5 ÷ 9 + 82 – 7
3)
100 ÷ 4 × 5 + 10
Validation (optional)
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
594
Expression
4)
7 × 8 – 5 + 62
5)
–6 (9) – (12 – 8) ÷ 2
6)
–3 (4 + 6)2 – 5 (3 – 8)2
7)
–2 (5.2 – 1.3) + (2.2)2 – 1
8)
(7.1)2 – (19.1 + 25.9) + 2 (0.2) – 12.3
Validation (optional)
Section 6.3 — Evaluating Expressions Using the Order of Operations
Expression
2
9)
10)
11)
12)
⎛ 3 1 ⎞ ⎛2 ⎞
5
÷ ⎜⎜ + ⎟⎟⎟ − ⎜⎜ ⎟⎟⎟
12 ⎜⎝ 8 4 ⎠ ⎜⎝ 3 ⎠
16 − 3 (7)
−8 + 3 (−4)
15 − 32
−14 − 2 (−1)
14 − 5 (−2)
3 (−1) − 10
595
Validation (optional)
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
596
TEAM EXERCISES
1. Simplify: 8 + 4 – 3 + 2 × 12 – 18 – 52
2. Change the same sequence of numbers and signs by adding one set of parentheses so that the simplified
answer to your new expression is –28.
8 + 4 – 3 + 2 × 12 – 18 – 52
3. Devise a real-life problem that would require applying the Order of Operations to solve.
IDENTIFY
AND
CORRECT
THE
ERRORS
Identify the error(s) in the following worked solutions. If the worked solution is correct, write “Correct” in the
second column. If the worked solution is incorrect, solve the problem correctly in the third column. You can
validate your work in the fourth column.
Worked Solution
What is Wrong Here?
1) Simplify:
2 – 3 (5 + 4)
Identify the Errors
Correct Process
Did not follow order
of operations
P, E, M&D, A&S.
2 − 3(5 + 4 )
= 2 − [3(5 + 4 )]
= 2 − [3(9)] P
Added 2 and –3
before multiplying.
The two terms are
2 and 3(5+4).
M
= 2 − 27
= 2 + (−27) A
= −25
Answer: –25
Validation
(optional)
Order P, M, A 9
9 −4 = 5 9
27 ÷ 9 = 3 9
−25 + 27 = +2 9
Section 6.3 — Evaluating Expressions Using the Order of Operations
Worked Solution
What is Wrong Here?
2) Simplify:
–24 ÷ (–3) (–4)
3) Simplify:
–32 × 2 –(16 – 9) + 6 ÷ 3
4) Simplify:
2
−3 + 5 − (−2)
16 ÷ (−4) + 2
Identify the Errors
597
Correct Process
Validation
(optional)
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
598
Worked Solution
What is Wrong Here?
5) Simplify:
(0.04)2 + 6.3 – (15 – 4.7)
6) Simplify:
−2 (−7) − 3 (5) − 1
23 (3) + 5 (−6)
Identify the Errors
Correct Process
Validation
(optional)
Section 6.3 — Evaluating Expressions Using the Order of Operations
ADDITIONAL EXERCISES
Simplify each of the following expressions:
1.
15 – 10 ÷ 5 × 2
2.
–3 (5 – 9) – 5 (3 – 6)
3.
−3 ( 4 − 7 ) − 5 ( 7 − 2)
−5 − 2 − 1
4.
16 – 4(–2)2 – (8 – 10)2
5.
3 (4 – 6) + 2 (1 + 3)2 – 33 ÷ (14 – 11)
6.
(0.2)3 + 0.5 × (0.3 + 6.5)
7.
(2.1)2 – 7.5 + 3 (0.63 + 0.27)
8.
⎛ 2⎞
5 ⎜⎜ ⎟⎟⎟ − 32
⎜⎝ 5 ⎠
16 − (−5)
9.
−4 (2) + 18
−16 − 2 (7)
10.
26 −14 (1− 3)
50 + 5 (−4)
11.
7 ⎛⎜ 1 ⎞⎟ 1 1
+⎜ ⎟ × − ÷ 3
8 ⎜⎝ 2 ⎟⎠ 3 2
2
3
12. (–2)3 + (–6)2 × (5 – 8) ÷ (–2)
599