Order of Operations (PEMDAS / BODMAS): Rules, Examples and Common Mistakes
October 30, 2026 · 6 min read
Why multiplication doesn't always come before division, how exponents and negatives trip students up, and worked examples you can check.
The order of operations is the agreement that lets everyone get the same answer to 3 + 4 × 2. Without it, one person would say 14 and another 11. The agreed answer is 11, because multiplication comes before addition.
The order
- P / B — Parentheses (Brackets): work inside them first.
- E / O — Exponents (Orders): powers and roots.
- MD — Multiplication and Division: from left to right.
- AS — Addition and Subtraction: from left to right.
PEMDAS (common in the US) and BODMAS (common in the UK, India and many other countries) describe exactly the same rules — only the words differ.
The most common mistake: M before D
Because "M" comes before "D" in PEMDAS, many students think multiplication always happens before division. It doesn't. They are partners at the same level, done in the order they appear from left to right.
- 8 ÷ 2 × 4 → work left to right: 8 ÷ 2 = 4, then 4 × 4 = 16 (not 1).
- The same is true for addition and subtraction: 20 − 6 + 2 = 14 + 2 = 16 (not 12).
A helpful picture: write PEMDAS as P – E – MD – AS, with M/D and A/S on shared steps.
Worked examples
- 3 + 4² = 3 + 16 = 19 (exponent first).
- 2 × (3 + 5)² = 2 × 8² = 2 × 64 = 128.
- 18 − 12 ÷ 3 × 2 = 18 − 4 × 2 = 18 − 8 = 10.
- (6 + 4) ÷ (7 − 2) + 3 = 10 ÷ 5 + 3 = 2 + 3 = 5.
Exponents and negative numbers
Watch for −3². The exponent applies only to the 3, so −3² = −(3 × 3) = −9. To square negative three you need parentheses: (−3)² = 9. Calculators follow this rule too, which surprises many students.
Fraction bars are grouping symbols
In (12 + 8) ÷ (2 × 5) written as a fraction, the bar acts like parentheses around the top and the bottom. Work out 20 on top and 10 on the bottom, then divide: 2.
How to practise
- Rewrite after each step. Copy the whole expression with one operation done. It is slower, but it removes almost all mistakes.
- Underline the next step before calculating it.
- Estimate. If 2 × (3 + 5)² gives you 66, something has gone wrong — the square makes it much bigger.
Under the Common Core, students first use parentheses in expressions in Grade 5 (5.OA.A.1) and add whole-number exponents in Grade 6 (6.EE.A.1–2).
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