What Is the Order of Operations?

The order of operations is a globally agreed set of rules that tells you which part of a mathematical expression to calculate first. Without it, the expression 6 + 2 × 3 could mean two different things — and two people calculating it independently could get two different answers. The rules exist to make mathematics unambiguous and universal.

Breaking Down BEDMAS Letter by Letter

• B — Brackets: Everything inside brackets is calculated first, from the innermost pair outward. Example: (3 + 2) × 4 = 20, not 11.
• E — Exponents: Powers and roots are evaluated next. Example: 2² + 3 = 7 — the exponent applies to 2 only.
• D / M — Division and Multiplication: These have equal priority. When both appear, work left to right. Example: 12 ÷ 4 × 3 = 9, not 1.
• A / S — Addition and Subtraction: Lowest priority, also left to right. Example: 10 − 3 + 2 = 9, not 5.

Why Does This Order Exist?

The order reflects how mathematical operations relate to each other in terms of structural power. Multiplication is repeated addition, and exponentiation is repeated multiplication. Operations with greater structural complexity are resolved first, so that the simpler operations work on the correct values. This hierarchy lets us write compact expressions without brackets around every sub-calculation.

The Three Most Common Mistakes

• Treating D and M as separate levels. Division does not always come before multiplication. They are equal priority. 2 × 6 ÷ 3 = 4, not 9. Left to right is the rule.
• Same mistake with A and S. Addition does not always come before subtraction. 10 − 3 + 2 = 9, not 5. Work left to right.
• Forgetting nested brackets. In (2 + (3 × 4)), solve the inner brackets first: 3 × 4 = 12, then 2 + 12 = 14.

A Full Worked Example

Walk through 3 + 2² × (8 − 5) ÷ 3 step by step:

• Brackets first: (8 − 5) = 3 → expression becomes 3 + 2² × 3 ÷ 3
• Exponents: 2² = 4 → expression becomes 3 + 4 × 3 ÷ 3
• Multiplication and division left to right: 4 × 3 = 12, then 12 ÷ 3 = 4 → 3 + 4
• Addition last: 3 + 4 = 7

A Quick Self-Assessment

Try these without a calculator: (1) 3 + 4 × 2 (2) (3 + 4) × 2 (3) 20 ÷ 4 + 6 × 2 (4) 2³ − (4 + 1) × 2

Answers: 11, 14, 17, −2. If any surprised you, a few sessions of BEDMAS on Hard mode will fix that faster than any worksheet.