Order of Operations Explained: PEMDAS Rules with Examples

The order of operations tells you which part of a math expression to calculate first. The rule is PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Follow that order every time and you get the same correct answer a calculator would.

For example, 3 + 4 × 2 = 11, not 14, because multiplication happens before addition.

A fully worked PEMDAS example

Evaluate: 2 + 3 × (5 - 1)² ÷ 4

1. Parentheses first: 5 - 1 = 4. Expression becomes 2 + 3 × 4² ÷ 4.
2. Exponents next: 4² = 16. Expression becomes 2 + 3 × 16 ÷ 4.
3. Multiplication and division, left to right: 3 × 16 = 48, then 48 ÷ 4 = 12. Expression becomes 2 + 12.
4. Addition last: 2 + 12 = 14.

Answer: 14.

The PEMDAS steps in order

1. Parentheses (grouping symbols). Do everything inside ( ), [ ], or above/below a fraction bar first. Example: in (2 + 3) × 4, compute 2 + 3 = 5 before multiplying.
2. Exponents. Handle powers and roots next. Example: 2 × 3² = 2 × 9 = 18.
3. Multiplication and division, left to right. These are equal priority — do whichever comes first as you read left to right. Example: 12 ÷ 3 × 2 = 4 × 2 = 8 (not 2).
4. Addition and subtraction, left to right. Same rule: equal priority, read left to right. Example: 10 - 4 + 3 = 6 + 3 = 9 (not 3).
5. Check your answer. Re-read the original expression and confirm each step matched PEMDAS.

Another worked example with a negative number

Evaluate: -8 + 3 × 2² - (6 - 2)

1. Parentheses: 6 - 2 = 4. Expression becomes -8 + 3 × 2² - 4.
2. Exponents: 2² = 4. Expression becomes -8 + 3 × 4 - 4.
3. Multiplication: 3 × 4 = 12. Expression becomes -8 + 12 - 4.
4. Addition and subtraction, left to right: -8 + 12 = 4, then 4 - 4 = 0.

Answer: 0.

Why PEMDAS matters

Without a fixed order, the same expression could give different answers. PEMDAS is the shared convention that makes math unambiguous — every textbook, calculator, and standardized test uses it.

Common mistakes with PEMDAS

• Doing addition before multiplication. In 3 + 4 × 2, students often add first to get 14. The correct answer is 11 because multiplication comes first.
• Treating multiplication as always before division. M and D are equal priority. In 8 ÷ 2 × 4, work left to right: 4 × 4 = 16, not 8 ÷ 8 = 1.
• Forgetting to flip left-to-right for subtraction. 10 - 3 - 2 = 5, not 9. Subtraction is not associative.
• Applying an exponent to a negative sign incorrectly. -3² means -(3²) = -9, but (-3)² = 9. Parentheses change the meaning.
• Ignoring the fraction bar as a grouping symbol. In (2 + 4) / 3, evaluate the numerator first before dividing.

Once you can name each PEMDAS step, the next skill is applying it fluently to messier expressions — nested parentheses, negatives, and fractions. If you want a guided tutor that walks your student through each step and catches mistakes as they happen, sign up to try LernOS below and practice the order of operations one problem at a time.