Order of Operations (PEMDAS): Rules, Steps & Examples
Part of the Math Concepts guide for Grades 3 to 8.
What Is the Order of Operations?
The order of operations is a set of rules that tells us the correct sequence for solving a math expression with more than one operation. Without these rules, the same expression could give different answers depending on how you work through it. In the United States, students use the acronym PEMDAS to remember the order: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
PEMDAS: The Order of Operations
| Step | Operation | What to do |
|---|---|---|
| P | Parentheses | Solve everything inside ( ), [ ], or { } first |
| E | Exponents | Evaluate all powers and roots next |
| M/D | Multiplication and Division | Work left to right, whichever comes first |
| A/S | Addition and Subtraction | Work left to right, whichever comes first |
Step-by-Step PEMDAS Examples
Let us walk through five examples together, starting simple and building up to the full PEMDAS sequence. Take a moment to try each one before reading the solution.
Example 1: Basic (no parentheses)
Problem: 3 + 4 x 2
- There are no parentheses or exponents, so we skip those steps.
- Multiplication comes before addition, so we multiply first: 4 x 2 = 8
- Now we add: 3 + 8 = 11
Answer: 11. A lot of students get 14 here by adding 3 and 4 first. Remember, multiplication always comes before addition when there are no parentheses.
Example 2: With parentheses
Problem: (3 + 4) x 2
- Parentheses come first: 3 + 4 = 7
- Now multiply: 7 x 2 = 14
Answer: 14. Adding the parentheses completely changed the answer from 11 to 14. That is exactly what parentheses are designed to do.
Example 3: With exponents
Problem: 2 + 3² x 4
- No parentheses, so we move to exponents: 3² = 9
- Multiplication next: 9 x 4 = 36
- Addition last: 2 + 36 = 38
Answer: 38. Students sometimes add the 2 and 3 together before squaring. Make sure to handle the exponent before any multiplication or addition.
Example 4: Mixed operations
Problem: 18 ÷ (2 + 1) − 3²
- Parentheses first: 2 + 1 = 3
- Exponents next: 3² = 9
- Division: 18 ÷ 3 = 6
- Subtraction last: 6 − 9 = −3
Answer: −3. Negative answers are perfectly valid here. Work through each step carefully and you will get there.
Example 5: Full PEMDAS
Problem: 5 x (2 + 3)² − 4 ÷ 2
- Parentheses: 2 + 3 = 5
- Exponents: 5² = 25
- Multiplication: 5 x 25 = 125
- Division: 4 ÷ 2 = 2
- Subtraction: 125 − 2 = 123
Answer: 123. This one uses every step. If you followed along and got 123, you have mastered the full order of operations.
Common Mistakes to Avoid
These are the three errors I see most often in the classroom. Read through each one carefully so they do not catch you off guard on a test.
Mistake 1: Treating multiplication and division as if multiplication always wins
Multiplication and division are equal in priority. You do not always multiply before you divide. Instead, you work left to right and handle whichever operation comes first.
Incorrect: 12 ÷ 4 x 3. Some students divide 4 x 3 first to get 12, then divide: 12 ÷ 12 = 1
Correct: Start from the left. 12 ÷ 4 = 3, then 3 x 3 = 9
Mistake 2: Applying an exponent to a negative sign
In the expression −3², the exponent only applies to the 3, not to the negative sign. So −3² = −9. If you want to square the negative number, you need parentheses: (−3)² = 9. This trips up a lot of students, so look out for it.
Mistake 3: Working nested parentheses from the outside in
When you have parentheses inside brackets, always start with the innermost set and work your way out.
For 4 x [2 + (3 − 1)]: start inside the round brackets first. (3 − 1) = 2. Then work the square brackets: [2 + 2] = 4. Then multiply: 4 x 4 = 16.
Common Questions About Order of Operations
What is PEMDAS?
PEMDAS stands for Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). It is a helpful acronym students use to remember the order of operations, which is the set of rules for solving math expressions in the correct sequence.
What is BODMAS?
BODMAS is the version of PEMDAS used in the United Kingdom and Australia. It stands for Brackets, Orders (which means exponents), Division and Multiplication, Addition and Subtraction. Both acronyms describe the exact same mathematical rules, just with slightly different wording.
Why do we need the order of operations?
Without an agreed set of rules, the expression 2 + 3 x 4 could equal 20 if you add first, or 14 if you multiply first. That kind of confusion would make math unreliable. The order of operations makes sure that every person solving the same expression gets the same answer.
Does multiplication always come before division?
No, and this is one of the most common misconceptions students have. Multiplication and division have equal priority. When both appear in an expression, you solve them from left to right, starting with whichever operation appears first. The same rule applies to addition and subtraction.
What grade do students learn PEMDAS?
In the United States, the order of operations is typically introduced in 5th or 6th grade. It aligns with Common Core standards for writing and evaluating numerical expressions, and students continue using it throughout middle school and high school math.
How do parentheses change the order of operations?
Parentheses override all other rules. Whatever is inside parentheses always gets solved first, no matter what. That is why (2 + 3) x 4 = 20, while 2 + 3 x 4 = 14. When you want a different order than the default, parentheses are the tool for the job.
