Basic Mathematics for College Students

308 Chapter 3 Summary and Review

SECTION 3.7 Order of Operations and Complex Fractions

Order of Operations

1. Perform all calculations within parentheses
   and other grouping symbols following the
   order listed in Steps 2–4 below, working
   from the innermost pair of grouping
   symbols to the outermost pair.


2. Evaluate all exponential expressions.


3. Perform all multiplications and divisions
   as they occur from left to right.


4. Perform all additions and subtractions as
   they occur from left to right.
   When grouping symbols have been removed,
   repeat Steps 2–4 to complete the calculation.
   If a fraction bar is present, evaluate the
   expression above the bar (called the
   numerator) and the expression below the bar
   (called the denominator) separately. Then
   perform the division indicated by the fraction
   bar, if possible.

DEFINITIONS AND CONCEPTS EXAMPLES

Evaluate:

First, we perform the subtraction within the second set of parentheses.

(There is no operation to perform within the first set.)

Evaluate the exponential expression:

.

Use the rule for dividing fractions: Multiply the first

fraction by the reciprocal of , which is.

Multiply the remaining factors

in the numerator.

Multiply the remaining factors

in the denominator.



4

15

To simplify, factor 12 as 3 4

and 9 as 33. Then remove the

common factor of 3 from the

numerator and denominator.



1  3

1

 4

3

1

 3  5

Multiply the numerators.

 Multiply the denominators.

1  12

9  5

12

5

5

 12

1

9



12

5

1312

2

 31  31  91



1

9



5

12

Subtract the numerators: 9 – 4 5.

Write the difference over the common

denominator 12.

5

12

a

1

3

b

2



Multiply the numerators.

b Multiply the denominators.

9

12



4

12

a

1

3

b

2

a

Within the parentheses, build

each fraction so that its

denominator is the LCD 12.

b

3

4



3

3



1

3



4

4

a

1

3

b

2

a

b

3

4



1

3

a

1

3

b

2

a

a

1

3

b

2

a

3

4



1

3

b

To evaluate a formula,we replace its variables

(letters) with specific numbers and evaluate the

right side using the order of operations rule.

Evaluate: for , , and.

This is the given formula.

Replace h, a,and bwith the given values.

Do the addition within the parentheses.

To prepare to multiply fractions, write as an

improper fraction and 4 as.

Write the improper fraction as a

mixed number by dividing 28 by 5.

28

 5 3 5

5

Multiply the remaining factors in the

numerator. Multiply the remaining

factors in the denominator.



28

5

To simplify, factor 4 as 22. Then

remove the common factor of 2 from

the numerator and denominator.



1  14  2

1

 2

2

1

 5  1

Multiply the numerators.

 Multiply the denominators.

1  14  4

2  5  1

4

1

(^2 45)


1

2

a

14

5

ba

4

1

b



1

2

a 2

4

5

b(4)



1

2

a 2

4

5

ba 1

1

3

 2

2

3

b

A

1

2

h (ab)

h 2

4

5

b 2

2

3

a 1

1

3

A

1

2

h(ab)