Order of Operations: Step-by-Step Guide
When a problem has more than one operation, the order you work in matters. PEMDAS is a way to remember the order: Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right). If everyone follows the same order, everyone gets the same answer.
Math Racer skill guide · Elementary · Grades K-5
How to solve it, step by step
- Do the parentheses first. Work out whatever is inside parentheses and brackets first. If there are parentheses inside parentheses, start with the innermost ones. A fraction bar also works like parentheses.\[(\,\text{innermost group}\,) \to \text{simplify first}\]
- Do the exponents next. After the parentheses are done, work out any powers, like 3 squared, and any square roots.\[a^{n},\;\sqrt[n]{a}\]
- Multiply and divide, left to right. Multiplication and division are equally important. Do whichever one comes first as you read from left to right.\[a \times b \div c \;=\; (a \times b) \div c\]
- Add and subtract, left to right. Last, do the addition and subtraction. They are also equally important, so go from left to right.\[a + b - c \;=\; (a + b) - c\]
Worked example
Evaluate: 3 + 6 x (5 + 4) / 3 - 7
Step 1: Parentheses first: (5 + 4) = 9. The problem is now 3 + 6 x 9 / 3 - 7.
Step 2: There are no exponents, so we move on.
Step 3: Multiply and divide from left to right: 6 x 9 = 54, then 54 / 3 = 18. The problem is now 3 + 18 - 7.
Step 4: Add and subtract from left to right: 3 + 18 = 21, then 21 - 7 = 14.
Step 5: The answer is 14.
Check: Go through the steps again slowly. 3 + 6(9)/3 - 7 = 3 + 54/3 - 7 = 3 + 18 - 7 = 14.
Tips and common mistakes
- Multiplication and division are equal partners. Do the one that comes first, reading left to right.
- Addition and subtraction are equal partners too. Go from left to right.
- Be careful with negatives and exponents. -3^2 means -(3 x 3) = -9, but (-3)^2 means (-3) x (-3) = 9.
- A fraction bar works like parentheses. Work out the top and the bottom first, then divide.
- 2(3 + 1) means 2 times (3 + 1). Do the parentheses first, then multiply.
- If you are not sure, add extra parentheses to show which part to do first.
- With parentheses inside parentheses, start from the inside and work out.