Something is wrong. The first part of the response (No price too high!) says to
buy the watch at any price. The second part (No! Price too high.) says not to buy it,
because it’s too expensive. The placement of the exclamation point makes us read
the two parts of the response differently, resulting in different meanings. When
reading a mathematical statement, the same kind of confusion is possible. For
example, consider the expressionWe can evaluate this expression in two ways. We can add first, and then multiply.
Or we can multiply first, and then add. However, the results are different.
Add 2 and 3 first.
30 Multiply 5 and 6.2 3 6 5 6
2 3 6
102 Chapter 1 Whole Numbers
Multiply 3 and 6 first.
20 Add 2 and 18.2 3 6 2 18
Different results
If we don’t establish a uniform order of operations, the expression has two
different values. To avoid this possibility, we will always use the following order of
operations rule. Order of Operations- 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. - Evaluate all exponential expressions.
- Perform all multiplications and divisions as they occur from left to right.
- 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.
It isn’t necessary to apply all of these steps in every problem. For example, the
expression does not contain any parentheses, and there are no exponential
expressions. So we look for multiplications and divisions to perform and proceed as
follows:
Do the multiplication first.
20 Do the addition.2 3 6 2 18
2 3 6
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.
Solution
Since the expression does not contain any parentheses, we begin with Step 2 of
the order of operations rule: Evaluate all exponential expressions. We will write
the steps of the solution in horizontal form.
2 42 8
Self Check 1
Evaluate:
Now TryProblem 19