Work out brackets first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right. Rewrite the whole expression after each step so nothing is carried in your head.
The rewrite method
Take a worked example: 8 + 6 ÷ (5 − 2) × 3.
- Brackets: 5 − 2 = 3, so the line becomes 8 + 6 ÷ 3 × 3.
- Division and multiplication, left to right: 6 ÷ 3 = 2, so 8 + 2 × 3.
- Then 2 × 3 = 6, so 8 + 6.
- Addition: 14.
Each line is a full expression, and you only change one thing per line. That is what stops the quiet slips.
Powers sit above multiplication
Try 12 − 3 × 2². The power comes first: 2² = 4. Then 3 × 4 = 12, and 12 − 12 = 0.
A student who subtracts first gets 9 × 4 = 36, which is very different. The order is not a preference, it is the agreed meaning of the expression.
The negative-number trap
Brackets decide what a power applies to. Compare the two:
| Expression | Working | Answer |
|---|---|---|
| −5² | −(5 × 5) | −25 |
| (−5)² | (−5) × (−5) | 25 |
Typing −5² into some calculators gives −25, and typing (−5)² gives 25. Knowing which one the question means matters before you press anything.
Left to right on equal rank
The instruction “multiplication and division, left to right” catches students in the middle of a line. Compare 24 ÷ 4 × 3 worked correctly and wrongly.
Correct: 24 ÷ 4 = 6, then 6 × 3 = 18. Wrong: 4 × 3 = 12 first, then 24 ÷ 12 = 2. The wrong route gives a completely different value, and only the first follows the rule.
Try it yourself
Work out 5 + 4 × (9 − 6)² ÷ 3.
Answer
Brackets: 9 − 6 = 3, so the line is 5 + 4 × 3² ÷ 3. Power: 3² = 9, so 5 + 4 × 9 ÷ 3.
Left to right: 4 × 9 = 36, then 36 ÷ 3 = 12. Finally 5 + 12 = 17. If you got 27, you added 5 and 4 before multiplying. Multiplication comes before addition.
Where to go next
Finish the cluster with checking a numerical result with estimation, which catches slips like these. If powers are the shaky part, try interpreting powers and roots. One-to-one help is available through SPM Mathematics tuition.