Exponents and Simplifying Expressions

Order of Operations

Motivation Problem

Find the value of   5 + 6(4).

Unless we have specific rules for finding the above value, different people could obtain different answers. Some may multiply first then add to obtain 29 and another person might add first then multiple to obtain 44. This situation is unacceptable . So, people have agreed on certain standard rules for determining the value of expressions that involve different operations. The most common rules and the ones that we are going to use are called the algebraic order of operations. Order of operations - Wikipedia, the free encyclopedia

We have learned the operations involving exponents, division, multiplication, subtraction and addition. In order to perform complex computations with these operations properly, we need to perform these operations in a particular order. The standard rules for the algebraic order of operations are:

First, we perform operations that are grouped such as by Parentheses.
Second, we compute Exponents.
Third, we perform the Multiplication and Division from left to right.
Finally, we perform the Addition and Subtraction from left to right.

The acronym for remembering the order of operations is PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition and Subtraction. A mnemonic device for remembering this acronym is Please E xcuse My Dear Aunt Sally.  

Example: The opening problem 5 + 6(4) would be worked as follows.

5 + 6(4) = 5 + 24  since multiplication comes before addition

            = 29

 A common mistake associated with this acronym is to forget that PEMDAS does not properly reflect the left-to-right part of the rules in the first paragraph. Multiplication and division have the same priority, and are done left-to-right. Addition and subtraction have the same priority, and are done left-to-right. A more accurate mnemonic might be   PE MD   AS to remind yourself that the underlined pairs are done together, moving left-to-right.

That is, multiplication and division are done in the same step, and they are done left-to-right.  

Example: In 100 ÷ 4 × 5, the multiplication and division must be done left-to-right, which means that in this case the division is actually done before the multiplication.

100 ÷ 4 × 5 = 25 × 5 = 125

In the same way, addition and subtraction are done in the same step, and they are also done left-to-right.  

Example: In 20 – 2 + 8, the addition and subtraction must be done left-to-right, which means in this case the subtraction is actually done before the addition:  

20 – 2 + 8 = 18 + 8 = 26

Examples: In each of the following the step to be done next is underlined.

Self-Check Problems

Evaluate each expression

92 – 7 · 9 Solution

7² + 4(3³ + 2) – 19 Solution

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