6.3 SIMPLIFYING EXPRESSIONS USING THE ORDER OF OPERATIONS
Brian has decided to open a savings account, where he will earn 4% interest, compounded semi-annually. He
will deposit $500 to open the account. He plans to leave the account alone for 1 year, making no additional
deposits and no withdrawals. At the end of that year, he will withdraw only the interest he has earned.
That amount can be determined by evaluating the following expression:
$500 × (1 + 0.04 ÷ 2)2 – $500
Complete the
withdrawal slip
at right, filling in
the amount Brian
will withdraw
from his account
in one year.
Assess your readiness to complete this activity. Rate how well you understand:
20.20
Not
ready
Almost
ready
Bring
it on!
• how to apply the Order of Operations to simplify an expression with
multiple operations
• the purpose for an agreed upon Order of Operations
• validation techniques for Order of Operations
•
Simplifying expressions
– accuracy
– documentation of steps
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Chapter 6 — Signed Numbers, Exponents, and Order of Operations
Order of Operations
To simplify an expression, the process must follow the Order of Operations:
First, simplify all operations within Parentheses.
Then, simplify all factors and terms with Exponents.
Then, compute Multiplication and Division, left to right as they occur in each term.
Finally, compute Addition and Subtraction, left to right as they occur in the expression.
While Example 1 is worked out, step by step, you are welcome to complete Example 2 as a running problem.
Space has been left for you to do precisely that.
Example 1: Simplify: 75 ÷ 5 × 2 + 23 − 4 (11 − 8)2
Try It!
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.
Example 2
Example 1
[(7 + 2) 2] + [5 ×9] −
[ 4 2] +[12 ÷ 4]
[75÷5×2]+[2 3]–[4(11–8)2]
The Expression is Written as a Fraction
Special
with an Expression in Either or Both the
Case:
Numerator and Denominator (see Model 4)
Step 2
Simplify
operations in
parentheses.
Simplify the operation(s)
within Parentheses, if
there are any, for each
term.
As each term is simplified
to one number, you
may drop the brackets
surrounding it.
To ensure that you are doing
the steps in the correct order
of operations, it may be
helpful to label each step as
you compute it.
266
[75÷5×2]+[2 3]–[4(11–8) 2]
=[75÷5×2]+[2 3]–[4(3) 2] P
P
[(9 ) 2] +[5 ×9] −
[ 4 2] +[12 ÷ 4]
Activity 6.3 — Simplifying Expressions Using the Order of Operations
Example 2
Steps in the Methodology
Example 1
Simplify the numbers with
Exponents, if there are
Simplify
numbers with any, in each term.
exponents.
=[75÷5×2]+[23]–[4(3)2]
Step 4
Compute Multiplication
and Division, left to right
as they are situated in
each term.
=[75÷5×2]+ 8 –[4×9]
Compute Addition and
Subtraction of the
simplified terms, left to
right as they are situated
in the expression.
= 30+ 8 – 36
Step 3
Multiply and
Divide left to
right.
Step 5
Add and
Subtract left
to right.
2×2×2
[ 81] +[5 ×9] −
E
[16] +[12 ÷ 4]
3×3
=[75÷5×2]+ 8 –[4×9] E
M&D
[ 81] +[ 45] −
[16] +[ 3]
=[15×2]+ 8 –[4×9]
=[30]+ 8 –[36] M & D
= 38 – 36
= 2
A & S
A&S
81 + 45 − 16 + 3
126 − 16 + 3
110 + 3
113
Step 6
Present your final answer.
Present the
answer.
Note: If the answer is in
fraction form, reduce it.
2
113
Model 1
Simplify
4.6 ÷ 2 – (0.5)2 + 2 (8 – 1.5)
Validation:
Step 1: Identify
the terms:
Step 2: P
[4.6 ÷ 2]
– [(0.5)2] + [2 (8 – 1.5)]
= [4.6 ÷ 2] – [(0.5)2] + [2 (6.5)]
0.5 × 0.5
Step 3: E
= [4.6 ÷ 2] –
Step 4: M & D
=
Step 5: A & S
= 2.05 + 13
= 15.05
left to right
as they occur
Step 6:
2.3 –
[0.25]
0.25
of Step 2
6.5 + 1.5 = 8
+ [2 (6.5)]
+ 13
of Step 4
2.3 × 2 = 4.6
13.0 ÷ 6.5 = 2
of Step 5
15.05 – 13 + 0.25
= 2.05 + 0.25 = 2.30
Order: P, E, M&D, A&S
Answer: 15.05
267
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
Model 2
Simplify
5 (–4) – (6 – 1)3 – 7 (–2)
Validation:
Step 1: Identify
the terms:
[5(–4)] – [(6 – 1)3] – [7 (–2)]
Step 2: P
= [5(–4)] –
[(5)3]
– [7 (–2)]
5×5×5
Step 3: E
= [5(–4)] –
125
– [7 (–2)]
Step 4: M
=
125
– (–14)
Step 5: S
= –20 + (–125) + (+14)
= –145 + (+14)
= –131
change all
subtraction to
addition
Step 6:
–20 –
of Step 2
5+1=6
of Step 4
–20 ÷ (–4) = +5
–14 ÷ (–2) = +7
of Step 5
–131 + (–14) + 125
= –145 + 125 = –20
Order: P, E, M, S
Answer: –131
Model 3
2
⎛ 1 ⎟⎞
⎜⎜ ⎟ + 2 − 3 × 10
⎝⎜ 2 ⎟⎠
3 5
Simplify
Validation:
⎡⎛ 1 ⎞2 ⎤
⎢⎜ ⎟⎟ ⎥ + ⎡⎢ 2 ⎤⎥ − ⎡⎢ 3 × 10⎤⎥
⎢⎜⎜⎝ 2 ⎟⎠ ⎥
⎥⎦
⎢⎣ 5
⎢⎣ 3 ⎥⎦
⎢⎣
⎥⎦
Step 1:
Step 2:
skip this step—no operations inside Parentheses
1 1
×
2 2
1
4
Step 3: E
=
Step 4: M
1
=
4
of Step 4
⎤
⎡3
⎡2⎤
+ ⎢ ⎥ − ⎢ × 10⎥
⎥⎦
⎢⎣ 5
⎢⎣ 3 ⎥⎦
⎡2⎤
+ ⎢ ⎥ −
⎢⎣ 3 ⎥⎦
3
2
10 6
×
= =6
1
1
1
5
Step 6:
268
=
6
11
1
− 6 = −5
12
12
Answer: – 5
1
12
6
3
=
10 5
of Step 5
–5
Step 5: A & S, left to right as they occur
1 2
3
8
11
+ =
+ =
4 3 12 12 12
6 ÷ 10 =
11 72 11 ⎛⎜ 72 ⎞⎟
− =
+ ⎜− ⎟
12 12 12 ⎜⎝ 12 ⎟⎠
11 + (−72)
61
1
=
= − = −5
12
12
12
1
11
+6 = +
12
12
11 2 11
8
− =
−
12 3 12 12
3
1
=
=
12
4
Order: E, M, A & S
Activity 6.3 — Simplifying Expressions Using the Order of Operations
Model 4
Simplify
Special Case: The Expression is Written as a Fraction with an Expression
in Either or Both the Numerator and Denominator
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.
Validation:
⎡4 (2 –7)⎤ − ⎡2 (5 –2)⎤
⎣
⎦
⎣
⎦
−10 − 2 − 1
Step 1:
⎡4 (– 5)⎤ −
⎦
= ⎣
−10 − 2
Step 2: P
of Step 2
–5 + 7 = 2
3+2=5
⎡2 (3)⎤
⎣
⎦
− 1
of Step 4
–20 ÷ (–5) = +4
6÷3=2
Step 3: no Exponents, skip this step
=
Step 4: M
=
Step 5: S
change to addition
Step 6: Reduce:
−20 − 6
−10 − 2 − 1
of Step 5
−20 + (−6)
−10 +
−26
= +2
−13
(−2)
+
(−1)
=
−26
−13
–26 + 6 = –20
–13 + 1 + 2
= –12 + 2 = –10
Order: P, M, S
Answer: 2
Validation and 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.
Models 1 through 4 demonstrate that keeping track of steps by labeling each one as you go is an
effective way to ensure that the order is correct. Validating the accuracy of each computation as you
work through the problem (as demonstrated in the models) can further ensure the accuracy of your
final answer.
269
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
Make Your Own Model
Either individually or as a team exercise, create a model demonstrating
how to solve the most difficult problem you can think of.
Answers will vary.
Problem: _________________________________________________________________________
1. What is the order of operations to follow when simplifying an expression?
PEMDAS: Perform operations in parenthesis first; then do all exponents; then do all multiplications or divisions
from left to right; then do all additions or subtractions, from left to right.
2. What is the purpose of the Order of Operations?
It serves as a standard to evaluate an expression and to simplify arriving at an answer.
270
Activity 6.3 — Simplifying Expressions Using the Order of Operations
3. How do you identify the terms of an expression? Give an example of an expression with four terms.
The terms of an expression are separated by addition (+) and subtraction (–) signs. Every expression has at least
one term.
4. When computing a series of multiplication and division operations within a single term, in what order must
they be done?
Multiplications and divisions must be done in order from left to right as they come in order. Divisions are done first
if they are the first operations beginning at the left of the expression to be evaluated.
5. In the final step of the Order of Operations, why must addition and subtraction be computed from left to
right as they occur in the expression?
The operation which comes first when working from left to right is what is to be done first. It could be the subtraction
that is done first. This is done this way because that is the way it is stated in the Order of Operations.
6. What is a strategy you can use to validate an order of operations problem?
It is easier to validate each step as you do it.
7. Why do you think the Exponents must be computed before the Multiplication and Division step is
computed?
This is universally accepted as the Order of Operations. Probably parentheses and exponents are done first because
they are more complicated to work out and more than one process is usually involved.
8. What aspect of the model you created is the most difficult to explain to someone else? Explain why.
Answers will vary.
271
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
Simplify each of the following expressions:
Expression
1) 16 – 23 ÷ 4 (2)
2) 32 × 5 ÷ 9 + 82 – 7
3) 100 ÷ 4 × 5 + 10
4) –2 (5.2 – 1.3) + (2.2)2 – 1
5) (7.1)2 – (19.1 + 25.9) + 2 (0.2) – 12.3
272
Validation (optional)
Activity 6.3 — Simplifying Expressions Using the Order of Operations
Expression
Validation (optional)
2
⎛ 3 1 ⎞ ⎛2 ⎞
5
÷ ⎜⎜ + ⎟⎟ − ⎜⎜ ⎟⎟
6)
12 ⎜⎝ 8 4 ⎟⎠ ⎜⎝ 3 ⎟⎠
7)
8)
9)
15 − 32
−14 − 2 (−1)
16 − 3 (7)
−8 + 3 (−4)
14 − 5 (−2)
3 (−1) − 10
Simplify each of the following expressions:
1. 15 – 10 ÷ 5 × 2
11
2. –3 (5 – 9) – 5 (3 – 6)
5.
27
3. (0.2) + 0.5 × (0.3 + 6.5)
3
3.408
4. (2.1) – 7.5 + 3 (0.63 + 0.27)
2
–4
6.
−3 ( 4 − 7 ) − 5 ( 7 − 2)
−5 − 2 − 1
⎛ 2⎞
5 ⎜⎜ ⎟⎟⎟ − 32
⎜⎝ 5 ⎠
16 − (−5)
2
2
26 −14 (1− 3)
7.
50 + 5 (−4)
3
−1
3
–1
7 ⎛1⎞ 1 1
8. + ⎜⎜⎜ ⎟⎟⎟ × − ÷ 3
8 ⎝ 2⎠ 3 2
3
4
273
Chapter 6 — Signed Numbers, Exponents, and Order of Operations
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.
Added 2 and –3 before
multiplying.
The two terms are
2 and 3(5+4).
2 - 3(5 + 4)
= 2 - [3(5 + 4)]
= 2 - [3(9)] P
= 2 - 27 M
= 2 + (-27) A
= -25
Answer: –25
Work all
multiplications OR
divisions working
from left to right.
2) Simplify:
–24 ÷ (–3) (–4)
2
3) Simplify:
−3 + 5 − (−2)
16 ÷ (−4) + 2
Perform order
of operations
PEMDAS.
Exponents should
be done first.
274
Validation
(optional)
Order P, M, A
9-4= 5
27 ÷ 9 = 3
-25 + 27 = +2
Activity 6.3 — Simplifying Expressions Using the Order of Operations
Worked Solution
What is Wrong Here?
4) Simplify:
(0.04)2 + 6.3 – (15 – 4.7)
Identify the Errors
Correct Process
Validation
(optional)
The decimal
point is placed
incorrectly in the
product of (.04)2.
Line up the
decimal points
and trailing zeros
when subtracting
4.7 from 15.
5) Simplify:
−2 (−7) − 3 (5) − 1
23 (3) + 5 (−6)
There needs to
be a single sign in
the final answer.
275