Basic Mathematics for College Students

2.6 Order of Operations and Estimation 185

EXAMPLE (^5) Evaluate:
StrategyWe will begin by evaluating the expression that is within the
parentheses. Since it contains more than one operation, we will use the order of
operations rule to evaluate it. We will perform the multiplication first and then the
addition.
WHYBy the order of operations rule, we must perform all calculations within the
parentheses first following the order listed in Steps 2–4 of the rule.
Solution
Do the multiplication within the
parentheses: 7  2 14.
Do the addition within the
parentheses:  4  14 10.
Do the multiplication: 3(10) 30.
 15 Do the addition.

 15  30

 15 3(10)

 15 3( 4  7  2 ) 15 3( 4  14 )

 4  7  2

 15 3( 4  7 2)

Expressions can contain two or more pairs of grouping symbols. To evaluate the
following expression, we begin within the innermost pair of grouping symbols, the
parentheses. Then we work within the outermost pair, the brackets.

Innermost pair

Outermost pair

67 5[ 1 (28)^2 ]

EXAMPLE (^6) Evaluate:
StrategyWe will work within the parentheses first and then within the brackets.
Within each pair of grouping symbols, we will follow the order of operations
rule.
WHYWe must work from the innermostpair of grouping symbols to the outermost.
Solution
Do the subtraction within the parentheses:
2  8 6.
Evaluate the exponential expression within
the brackets.
Do the addition within the brackets:
 1  36 35.
Do the multiplication: 5(35) 175.
If it is helpful, use the subtraction rule:
Add the opposite of 175, which is 175.
  108 Do the addition.

 67 (175)

 67  175

 67 5[35]

 67 5[ 1 36]

 67 5[ 1 ( 6 )^2 ]

67 5[ 1 ( 2  8 )^2 ]

67 5[ 1 (28)^2 ]

17

6

5

15

 67

108

3

2

5

 5

175

Self Check 5

Evaluate:

Now TryProblem 29

 18 6( 7  9 2)

Self Check 6

Evaluate:

Now TryProblem 33

81 4[ 2 (59)^2 ]

 

 

Success Tip Any arithmetic steps that you cannot perform in your head

should be shown outside of the horizontal steps of your solution.