Order of Operations Lesson

The order of operations tells us which operation to perform in which order when faced with multiple operations:

1. Work inside of any parentheses or grouping symbols
2. Perform all exponent operations
3. Multiply or divide (working left to right)
4. Add or subtract (working left to right)

People often come up with tricks to remember the order of operations. You will often hear the term PEMDAS thrown around. This will also be attached to a phrase where each first letter corresponds to building PEMDAS. One such common phrase is:
Please Excuse My Dear Aunt Sally
When we break PEMDAS down, we see the following:

• P - Parentheses or grouping symbols
• E - Exponents
• M - Multiply
• D - Divide
• A - Addition
• S - Subtraction

This can be a bit misleading, as multiplication is listed before division and addition is listed before subtraction. This is not the correct order. Multiplication and Division have the same priority level and are evaluated working left to right. When working a problem, if division comes first working left to right, it has a higher priority and is executed first. The same is true with Addition and Subtraction, these operations have the same priority level and are executed working left to right. When using PEMDAS, modify the structure to:

• P - Parentheses or grouping symbols
• E - Exponents
• M/D - Multiply/Divide (working left to right)
• A/S - Addition/Subtraction (working left to right)

Let's take a look at a few problems:
Example 1: Evaluate each.
(5 + 3²) x 3 ÷ 7

• First, we look for parentheses or grouping symbols, once inside we will reapply the order of operations.
• (5 + 3²) x 3 ÷ 7
• Inside of the parentheses we have: 5 + 3², we have addition and an exponent operation. Since exponents have a higher priority, we will evaluate 3² first, and then add 5 to the result.
• 3² = 9, replace 3² with 9:
• (5 + 9) x 3 ÷ 7
• We still have one operation inside of parentheses. We will now add 5 and 9. 5 + 9 = 14, replace 5 + 9 with 14 and remove the parentheses.
• 14 x 3 ÷ 7
• Now, we have multiplication and division left. These have the same level of priority and are worked left to right. Since multiplication is to the left of the division, we multiply first, and then divide. 14 x 3 = 42, replace the 14 x 3 with 42.
• 42 ÷ 7
• Now, we have only division left. 42 ÷ 7 = 6, which is our final answer.

(5 + 3²) x 3 ÷ 7 = 6
Example 2: Evaluate each.
21 ÷ 7 x 3 + (2² x 3² + 4 - 39)

• First, we look for parentheses or grouping symbols, once inside we will reapply the order of operations.
• 21 ÷ 7 x 3 + (2² x 3² + 4 - 39)
• Inside of parentheses, we have exponent operations, multiplication, addition, and subtraction. Exponents have the highest priority. We will perform 2² and 3² first. 2² = 4, 3² = 9. Let's replace these in our problem.
• 21 ÷ 7 x 3 + (4 x 9 + 4 - 39)
• Inside of parentheses, we have multiplication, addition, and subtraction. The multiplication has the highest priority. 4 x 9 = 36, let's replace this in our problem.
• 21 ÷ 7 x 3 + (36 + 4 - 39)
• Inside of parentheses, we now have only addition and subtraction. These are evaluated from left to right. Addition occurs to the left of the subtraction. This means we will first find the sum of 36 and 4. 36 + 4 = 40, let's replace this in our problem.
• 21 ÷ 7 x 3 + (40 - 39)
• Inside of parentheses, we now have only subtraction. 40 - 39 = 1, let's replace this in our problem and remove the parentheses.
• 21 ÷ 7 x 3 + 1
• Now that the parentheses are dealt with, we have division, multiplication, and addition left in our problem. We know multiplication and division have a higher priority than addition. We multiply and divide from left to right, so 21 ÷ 7 will come first. 21 ÷ 7 = 3, let's replace this in our problem.
• 3 x 3 + 1
• Now, we have only multiplication and addition, multiplication has a higher priority. This means 3 x 3 will come first. 3 x 3 = 9, let's replace this in our problem.
• 9 + 1
• Our last step is to add 9 and 1. We know that 9 + 1 = 10, which is our final answer.

21 ÷ 7 x 3 + (2² x 3² + 4 - 39) = 10

---

Skills Check: