# The Order of Operations

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:

1. Parentheses
2. Exponents
3. Multiplication and Division
4. 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.