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Order of Operations

(Priority Rules for Arithmetic)

To ensure that even the most complicated mathematical expressions have the same meaning forevery reader, the following rule is always observed in evaluating an expression:

For all numerical or algebraic expressions, the order of evaluation is:

1. parentheses or brackets first (starting with the innermost)

2. exponentials or powers next3. multiplications and divisions next

4. additions and subtractions last

5. if an expression involves three or more operations at the same level of priority, those operations are done from left to right.

These rules are important. If you don’t obey them when evaluating numerical expressions, orwhen manipulating algebraic expressions, you will get results that the rest of the world considersto be incorrect.


Common Error #1:

Be careful with sequences of additions and subtractions. For example, if we proceeded asfollows:

5 - 3 + 6   5 - 9 Add the 3 to the 6 to get 9 – this violates thefifth part of the rule. Addition and subtractionare at the same priority level, so in this case weshould do the ‘-‘ between the 5 and the 3before doing the ‘+’ found further to the right.
    - 4 The wrong answer!
Instead, the rules require  
5 - 3 + 6   2 + 6 The subtract and add are at the same level ofpriority, so the subtract, being the leftmost ofthe two operations, is done first. Thus, 5 – 3gives 2
    8 Only one operation is left, so add 2 to 6 to get8, the correct answer!


Common Error #2:

Another common error is to overlook higher priority operations that may not be obviously present.This often happens when multiplications are overlooked, because no specific multiply operator ispresent. For example

3 + 5 (10 - 6)   8 (10 - 6) Add the 3 and the five to the left. This is anerror, because the 5 itself is to be multipliedonto the result of evaluating the bracketedexpression, and both the evaluation of thebracketed expression and the multiplication arehigher priority than the addition. The ‘+’between the 3 and 5 is the lowest priorityaccording to the rule, and yet was done firsthere in error
    8 (4) Evaluate the bracketed expression
    32 which is an incorrect final result.
Instead, the evaluation of this expression should have proceeded as follows:
      Evaluate the bracketed expression – this is thehighest priority operation present
      Do the multiplication – it has higher prioritythan the addition
      Finally, do the remaining addition, to get thecorrect final result of 23.
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