In a calculation with more than one operation, work in this order: brackets, then powers, then multiplication and division, then addition and subtraction. Operations of equal rank are done from left to right.
The rule is used in every part of number sense and exact arithmetic and returns whenever you substitute into a formula in algebra and mensuration.
Why does the order matter?
Take 2 + 3 × 4. If you add first, you get 5 × 4 = 20. If you multiply first, you get 2 + 12 = 14.
Only one answer can be right, so mathematicians agreed on a fixed order, and calculators follow it too.
Multiplication and division are “stronger” than addition and subtraction, so they bind tighter. Brackets override everything because they mark a part to be done first.
How to work through it step by step
- Brackets first. Work out the inside of each bracket, using the same order inside it.
- Powers and roots next. Square, cube or root what is required.
- Multiplication and division. Work through them from left to right, as they appear.
- Addition and subtraction. Again from left to right.
- Rewrite the whole line after each step. Never try to do two stages in your head.
Worked example
Work out 8 + 6 × (9 − 5)² ÷ 4.
Step 1, brackets: 9 − 5 = 4, so the line becomes 8 + 6 × 4² ÷ 4.
Step 2, powers: 4² = 16, giving 8 + 6 × 16 ÷ 4.
Step 3, multiply and divide from left to right: 6 × 16 = 96, then 96 ÷ 4 = 24, giving 8 + 24.
Step 4, add: 8 + 24 = 32.
Check: the power and the multiplication happen before the final 8 is added, so the answer must be larger than 8 and well above 24. 32 fits.
The mistake to watch for
A common slip is to work strictly from left to right, as in reading a sentence.
Mistaken answer: 8 + 6 × 16 ÷ 4 = 14 × 16 ÷ 4 = 224 ÷ 4 = 56
The student added 8 + 6 before multiplying.
Addition ranks below multiplication and division, so 8 + 6 must wait. Multiply and divide first, then add: 6 × 16 ÷ 4 = 24, and 8 + 24 = 32.
A second slip is to do the division before the multiplication when they appear in the other order. Always move left to right within the same rank.
Check yourself
Try these without a calculator, then open each answer.
1. Work out 20 − 12 ÷ 3 × 2.
Show answer
Division and multiplication first, left to right: 12 ÷ 3 = 4, then 4 × 2 = 8. So 20 − 8 = 12. (Doing 3 × 2 first would wrongly give 12 ÷ 6 = 2 and an answer of 18.)
2. Work out (−4)² − 3 × (2 − 5).
Show answer
Bracket: 2 − 5 = −3. Power: (−4)² = 16. Multiply: 3 × (−3) = −9. Then 16 − (−9) = 16 + 9 = 25.
3. Work out (7 + 5) / (4 − 1) + 18 ÷ (2 + 4) × 3.
Show answer
First part: 12 / 3 = 4. Second part: 18 ÷ 6 = 3, then 3 × 3 = 9. Total: 4 + 9 = 13.
Where this leads next
Put it together with the rest of the module in the number sense practice set, and review fractions in working with fractions without decimal rounding if a fraction step felt slow. The non-calculator working trainer lets you test each line.
Students often know the rule by name but skip it when the pressure is on. A teacher in online one-to-one Mathematics tuition can watch for that moment in your working and build a habit that holds in the exam.