When simplifying expressions, it is important that we simplify them in the correct order. Consider the following problem done two different ways:

Add First 


Multiply  

Multiply 


Add  

Solution 


Solution 
The previous example illustrates that if the same problem is done two different ways we will arrive at two different solutions. However, only one method can be correct. It turns out the second method, 17, is the correct method. The order of operations ends with the most basic of operations, addition (or subtraction). Before addition is completed we must do repeated addition or multiplication (or division). Before multiplication is completed we must do repeated multiplication or exponents. When we want to do something out of order and make it come first we will put it in parenthesis (or grouping symbols). This list then is our order of operations we will use to simplify expressions.
This format helps us remember that multiplication and division are done left to right (same with addition and subtraction). Another way students remember the order of operations is to think of a phrase such as “Please Excuse My Dear Aunt Sally” where each word starts with the same letters as the order of operations start with.

Parenthesis first  

Exponents  

Multiply  

Add  

Our Solution  
 

Divide first (left to right!)  

Multiply  

Our Solution 
In the previous example, if we had multiplied first, five would have been the answer which is incorrect. If there are several parenthesis in a problem we will start with the inner most parenthesis and work our way out. Inside each parenthesis we simplify using the order of operations as well. To make it easier to know which parenthesis goes with which parenthesis, different types of parenthesis will be used such as { } and [] and ( ), these parenthesis all mean the same thing, they are parenthesis and must be evaluated first.
Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html