The rule in one line
PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Some countries teach it as BODMAS or BIDMAS — the rule is identical.
The part almost everyone gets wrong
Multiplication does not come before division, and addition does not come before subtraction. Each pair sits on the same level and you work left to right.
Example: 12 ÷ 3 × 2. Left to right gives 4 × 2 = 8. If you did the multiplication first you would get 12 ÷ 6 = 2, which is wrong.
Example: 10 - 4 + 3. Left to right gives 6 + 3 = 9, not 10 - 7 = 3.
A worked example
Simplify 6 + 2 × (5 - 1)² ÷ 8.
Parentheses first: 5 - 1 = 4, so we have 6 + 2 × 4² ÷ 8. Exponent next: 4² = 16, giving 6 + 2 × 16 ÷ 8. Multiplication and division left to right: 2 × 16 = 32, then 32 ÷ 8 = 4, giving 6 + 4. Addition: 10.
Negatives and fraction bars
A fraction bar acts like a pair of invisible parentheses: simplify the whole top and the whole bottom before dividing. With -3², the exponent applies only to 3, so the answer is -9; (-3)² is 9. This single distinction accounts for a large share of lost marks.
Practice
Try: 8 - 3 × 2 + 5; 24 ÷ (2 + 4) × 3; 2 + 3² × (8 - 6); (7 - 2)² ÷ 5 - 1.
Answers: 7, 12, 20, 4.