Section 7.5
Order of Operations with
Real Numbers
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Objectives
o Use the rules for order of operations to evaluate
expressions.
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Using the Rules for Order of Operations
1.
2.
3.
4.
Rules for Order of Operations
Simplify within grouping symbols, such as
parentheses ( ), brackets [ ], and braces { },
working from the innermost grouping outward.
Find any powers indicated by exponents.
Moving from left to right, perform any
multiplications or divisions in the order they
appear.
Moving from left to right, perform any additions or
subtractions in the order they appear.
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Using the Rules for Order of Operations
Notes
Other grouping symbols are the absolute value bars
47
(such as 3  5 ), the fraction bar (as in
), and the
10  1
square root symbol (such as 5  11 ).
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Using the Rules for Order of Operations
Notes
Even though the mnemonic PEMDAS is helpful,
remember that multiplication and division are
performed as they appear, left to right. Also, addition
and subtraction are performed as they appear, left to
right.
For example
12  3  4  4  4  16,
but
12  3  4  36  4  9.
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Example 1: Order of Operations with Integers
Use the rules for order of operations to evaluate each
of the following expressions.
a. 36  4  6  22
Solution
36  4  6  2
 36  4  6  4
 9  24
 15
2
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exponents.
Divide and multiply, left to right.
Subtract.
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Example 1: Order of Operations with Integers
(cont.)


b. 2 32  1  3  23
Solution


2 32  1  3  23
 2  9  1  3  8
 2 8  3  8
 16  24
 8
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exponents
Subtract inside the parentheses.
Multiply.
Subtract (or add algebraically).
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Example 1: Order of Operations with Integers
(cont.)




c. 9  2  3  5  72  2  22 
Solution
9  2  3  5  72  2  22 
 9  2  3  5  49   2  4 
 9  2 15  49  2  4 
 9  2  34   2  4 
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exponents
Multiply inside the parentheses.
Subtract inside theparentheses.
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Example 1: Order of Operations with Integers
(cont.)
 9  2 17  4 
 9  2 13
 9  26
 35
Divide inside the brackets.
Add inside the brackets.
Multiply.
Add.
Note: Because of the rules for order of operations at no
time did we even subtract 9 − 2.
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Completion Example 2: Order of Operations
with Integers


Evaluate the expression 6 52  42  2  33.
Solution


6 5  4  23
2
2
3
27
25  ____
16   2  ____
 6  ____
exponents
9   2  27
 6  ____
54  ____
54
 ____
Multiply.
0
 ____
Subtract.
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Simplify within parentheses.
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Example 3: Order of Operations with Fractions
Use the rules for order of operations to evaluate each
of the following expression.
1  1 1
a. 2    
3  4 3
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Example 3: Order of Operations with Fractions
(cont.)
1  1 1
Solution 2    
3  4 3
7  3 4
   
3  12 12 
7  7
  
3  12 
Change the mixed number to
an improper fraction and find
the LCD in the parentheses.
Add within parentheses.
4
7 12
 
3 7
Divide and reduce.
4
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Example 3: Order of Operations with Fractions
(cont.)
3 5 1  5
b.     
5 6 4  2
Solution
2
3 5 1  5
   
5 6 4  2
2
3 5 1  25 
    
5 6 4  4
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exponents
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Example 3: Order of Operations with Fractions
(cont.)
3 5 1 4
   
5 6 4 25
Divide, then multiply and reduce.
1 1
 
2 25
25 2


50 50
27

50
Find the LCD.
2
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Add.
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Practice Problems
Use the rules for order of operations to evaluate each
expression.
1. 36  2.6
2. 8  2  4  2   42
3. 9  3   2  5 
2
 3 3 4 2 2
5.       
 4 8 5 5 3
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2
7 5
1
4.    
 6  12 8
1 1 
6. 15    
 3 10 
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Practice Problems (cont.)
Simplify.
2
16
2
b
7
7.
  
5a 3a 7 5b
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Practice Problem Answers
1. 38.6
2. 8
3. 27
97
4. 
168
13
7. 
40
2
5.
3
450
6. 
13
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