Consider the following:
6 + 3 * 2
There are two operations we can choose to do first, either 6 + 3 or 3 * 2. If we start with 6 + 3, then we get:
6 + 3 * 2
9 * 2
18
But if we start with 3 * 2, then we have:
6 + 3 * 2
6 + 6
12
And we’re getting two different answers! So how do we decide what operation to do first? The answer is by following the order of operations.
The order of operations is the following:
- Parentheses
- Exponents
- Multiplication and Division
- Addition and Subtraction
First, you do anything that is inside of parentheses. So if we have 2 * (4 + 2), then we will do 4 + 2 first since it’s inside of the parentheses.
Second, you deal with any exponents. So if we have 2 * 4², we will do 4² before we do 2 * 4.
Third, you do multiplication and division, starting on the left and going to the right. So if we have 2 * 12 / 3 + 2, first we’ll do 2 * 12 to get 24, and that leaves us with 24 / 3 + 2. Next we’ll do 24 / 3 to get 8. Finally, we’ll have 8 + 2, which is 10.
Fourth, you do any addition and subtraction, beginning on the left and working to the right. For example, if we have 3 + 2 – 4, we’ll do 3 + 2 first to get 5, then do 5 – 4 to get 1.
You can remember the order of operations by using the following pneumonic device:
Please Excuse My Dear Aunt Sally
Parentheses, Exponents, Multiplication & Division, Addition & Subtraction
Here are some more in-depth examples.
Example 1:
(3 + 2)² / 5 + 6
First we do what’s inside of the parentheses; 3 + 2 gives us 5.
5² / 5 + 6
Next we do the exponents, and 5² gives us 25.
25 / 5 + 6
Now we do the division 25 / 5 to get 5.
5 + 6
Finally we do the addition, and 5 + 6 is 11.
Example 2:
1 + 2 * 3 – 4 + 5 * 6 + 7 – 8 / 2
There are no parenthesis or exponents, so first we do the multiplication and division. 2 * 3 gives us 6.
1 + 6 – 4 + 5 * 6 + 7 – 8 / 2
From here, 5 * 6 gives us 30.
1 + 6 – 4 + 30 + 7 – 8 / 2
The only multiplication or division left is 8 / 2, which is 4.
1 + 6 – 4 + 30 + 7 – 4
Now we do addition and subtraction, starting from the left and working to the right. We know 1 + 6 is 7.
7 + 6 – 4 + 30 + 7 – 4
And 7 + 6 is 13.
13 – 4 + 30 + 7 – 4
We continue this as follows:
13 – 4 + 30 + 7 – 4
9 + 30 + 7 – 4
39 + 7 – 4
46 – 4
42
And we get the answer of 42.