Basic Mathematics for College Students

2.6 Order of Operations and Estimation 183

SECTION 2.6

Order of Operations and Estimation

In this chapter, we have discussed the rules for adding, subtracting, multiplying,
and dividing integers. Now we will use those rules in combination with the order of
operations rule from Section 1.9 to evaluate expressions involving more than one
operation.

Objectives

1 Use the order of operations rule.

2 Evaluate expressions containing

grouping symbols.

3 Evaluate expressions containing

absolute values.

4 Estimate the value of an

expression.

1 Use the order of operations rule.

Recall that if we don’t establish a uniform order of operations, an expression such as
can have more than one value. To avoid this possibility, always use the
following rule for the order of operations.

2  3  6

Order of Operations

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


2. Evaluate all the 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 (the denominator) separately.
   Then perform the division indicated by the fraction bar, if possible.

We can use this rule to evaluate expressions involving integers.

EXAMPLE (^1) Evaluate:
StrategyWe will scan the expression to determine what operations need to be
performed. Then we will perform those operations, one at a time, following the
order of operations rule.
WHYIf we don’t follow the correct order of operations, the expression can have
more than one value.
SolutionAlthough the expression contains parentheses, there are no calculations
to perform withinthem. We begin with step 2 of the order of operations rule:
Evaluate all exponential expressions.
Evaluate the exponential expression:
(3)^2 9.
Do the multiplication: 4(9) 36.
If it is helpful, use the subtraction rule:
Add the opposite of 2, which is 2.
 34 Do the addition.

 36  2

 36 (2)

4(3)^2 (2)4( 9 )(2)

4(3)^2 (2)

Self Check 1

Evaluate:

Now TryProblem 13

5(2)^2 (6)