Order of Operations with Brackets – BODMAS Explained
Order of operations is the single rule that makes arithmetic notation unambiguous. Without an agreed sequence for evaluating a string of operations, the expression 8 + 4 × 3 could mean either “add 8 and 4, then multiply by 3” (giving 36) or “multiply 4 by 3, then add 8” (giving 20) — and only one of those is accepted as correct. Mathematicians settled on a fixed hierarchy as printed algebra spread across Europe centuries ago, and by the twentieth century it had been distilled into classroom mnemonics: BODMAS in the UK and much of the Commonwealth, PEMDAS (often remembered as “Please Excuse My Dear Aunt Sally”) in the United States. Both describe exactly the same hierarchy — they simply group the letters differently.
Brackets sit at the very top of that hierarchy. They are the one tool that lets you override the default order and force a calculation to happen in whatever sequence you actually intend. Understanding brackets thoroughly — including nested brackets, brackets combined with powers, and the genuinely ambiguous cases that trip up even confident adults — is essential, because every calculator, spreadsheet formula, and programming language you will ever use evaluates expressions using this exact same set of rules.
The Complete BODMAS / PEMDAS Hierarchy
| Step | BODMAS | PEMDAS | What it covers |
|---|---|---|---|
| 1 | B – Brackets | P – Parentheses | ( ), [ ], { } — innermost pair first |
| 2 | O – Orders (powers/roots) | E – Exponents | Squares, cubes, other powers, and roots |
| 3 | D – Division | M – Multiplication | Equal precedence with the row below — left to right |
| 4 | M – Multiplication | D – Division | Equal precedence with the row above — left to right |
| 5 | A – Addition | A – Addition | Equal precedence with the row below — left to right |
| 6 | S – Subtraction | S – Subtraction | Equal precedence with the row above — left to right |
The most common BODMAS misconception: division does not always come before multiplication, and addition does not always come before subtraction. Despite the order the letters happen to be written in, D & M form a single equal-precedence pair, and so do A & S. When two operations from the same pair sit next to each other, you simply work through them from left to right — whichever one comes first in the expression is the one you do first.
Without vs With Brackets — Same Numbers, Different Answers
The next two calculations use exactly the same three numbers and the same two operations. The only difference is where a single pair of brackets sits — yet that is enough to change the answer completely.
Step 1 — multiplication ranks above addition, so it happens first: 4 × 3 = 12.
Step 2 — now add: 8 + 12 = 20.
Step 1 — brackets always go first, even though addition normally ranks below multiplication: 8 + 4 = 12.
Step 2 — now multiply: 12 × 3 = 36.
Types of Brackets and How They Nest
When a single pair of round brackets is not enough — because you need brackets inside brackets — mathematicians use three different bracket shapes so it is easy to see, at a glance, which opening bracket matches which closing one. The convention, from innermost to outermost, is:
| Symbol | Name | Typical position |
|---|---|---|
| ( ) | Round brackets / parentheses | Innermost |
| [ ] | Square brackets | Middle |
| { } | Curly brackets / braces | Outermost |
Regardless of shape, every bracket is treated identically by BODMAS: it simply means “evaluate what is inside me before doing anything with what is outside me.” When brackets are nested, always start with the innermost pair and work outward, one layer at a time.
Working Through Nested Brackets, Step by Step
Step 1 — innermost brackets: 3 + 4 = 7.
Step 2 — the expression is now 2 × [7 × 5 − 6]. Inside the square brackets, multiplication ranks above subtraction: 7 × 5 = 35.
Step 3 — the square brackets now read [35 − 6] = 29.
Step 4 — only the outer multiplication remains: 2 × 29 = 58.
Worked Examples – From Straightforward to Tricky
(9 − 3) ÷ 2 + 5
Step 1 — brackets first: 9 − 3 = 6.
Step 2 — division ranks above addition: 6 ÷ 2 = 3.
Step 3 — finally add: 3 + 5 = 8.
3 × (2 + 4²) − 10
Step 1 — inside the brackets, the power ranks above addition, so evaluate it first: 4² = 16.
Step 2 — now finish the bracket: 2 + 16 = 18.
Step 3 — multiply: 3 × 18 = 54.
Step 4 — subtract: 54 − 10 = 44.
4 × {2 + (3 × 5)} − 7
Step 1 — innermost round brackets first: 3 × 5 = 15.
Step 2 — the curly brackets now read {2 + 15} = 17.
Step 3 — multiply: 4 × 17 = 68.
Step 4 — subtract: 68 − 7 = 61.
40 ÷ 8 × 5
Division and multiplication have exactly equal precedence, so BODMAS does not mean “always divide before you multiply” — it means work through them in the order they are written, left to right.
Step 1 — the leftmost operation is division: 40 ÷ 8 = 5.
Step 2 — now multiply: 5 × 5 = 25.
Common mistake: multiplying first instead (8 × 5 = 40, then 40 ÷ 40 = 1) gives the wrong answer of 1, because it ignores the left-to-right rule and treats multiplication as if it always outranks division.
4² + (10 − 3 × 2) ÷ 4
Step 1 — brackets first; inside them, multiplication ranks above subtraction: 3 × 2 = 6.
Step 2 — finish the bracket: 10 − 6 = 4.
Step 3 — powers rank above division and addition, so evaluate 4² next: 4² = 16.
Step 4 — division: 4 ÷ 4 = 1.
Step 5 — finally add: 16 + 1 = 17.
8 ÷ 2(2 + 2)
This expression regularly goes viral online because it genuinely divides opinion. Using the modern, internationally taught convention — that “2(…)” written next to a bracket is ordinary multiplication with exactly the same precedence as any other multiplication — the calculation runs left to right through the division and multiplication once the bracket is resolved:
Step 1 — brackets first: 2 + 2 = 4.
Step 2 — read left to right: 8 ÷ 2 = 4.
Step 3 — then multiply: 4 × 4 = 16.
Some older textbooks and calculators instead treat “2(4)” as a single inseparable term and multiply it out before dividing, which would give 8 ÷ 8 = 1. Because two equally confident answers exist, this is exactly the kind of expression professional mathematicians avoid writing — they add extra brackets instead, either 8 ÷ (2 × (2 + 2)) = 1 or (8 ÷ 2) × (2 + 2) = 16, so nobody has to guess.
Common BODMAS Mistakes to Avoid
- Treating “D before M” and “A before S” literally. Division and multiplication are equal rank; so are addition and subtraction. Work left to right within each pair.
- Forgetting that brackets override everything. Even a power or a multiplication sitting outside a bracket must wait until the bracket’s contents are fully resolved.
- Evaluating powers after multiplication instead of before it. A term like 3 × 4² means 3 × 16 = 48, not 12² = 144.
- Losing track of which bracket closes which when brackets are nested. Always resolve the innermost pair first and work outward one layer at a time.
- Writing ambiguous notation. When in doubt, add an extra pair of brackets to make your intended order explicit — it costs nothing and removes all doubt.
Where Order of Operations Actually Gets Used
BODMAS is not just a school exercise — every device that performs a calculation on your behalf relies on exactly this hierarchy, and knowing it well helps you spot when a tool is misbehaving.
- Scientific and graphing calculators apply the full BODMAS hierarchy automatically, which is why typing 8 + 4 × 3 correctly returns 20.
- Basic four-function calculators often do not — they may evaluate strictly left to right with no operator precedence at all, so identical button presses can give a different (and mathematically incorrect) answer. This is a genuine, well-documented source of classroom confusion.
- Spreadsheet formulas in Excel or Google Sheets (for example =8+4*3) follow BODMAS/PEMDAS exactly, and brackets are used constantly to force a particular calculation order in financial and scientific formulas.
- Programming languages such as Python, JavaScript, and C all define an “operator precedence” table that is, at its core, the same BODMAS hierarchy extended to cover additional operators.
Key Takeaways
- Brackets are always evaluated first, and nested brackets are resolved from the innermost pair outward.
- Powers (orders/exponents) rank above multiplication, division, addition, and subtraction, but below brackets.
- Division and multiplication are equal precedence — read the expression left to right; the same is true for addition and subtraction.
- BODMAS and PEMDAS describe exactly the same hierarchy; they are simply different mnemonics used in different countries.
- Genuinely ambiguous notation exists — when it does, add brackets to remove all doubt rather than relying on convention alone.
- Every calculator, spreadsheet, and programming language you use evaluates expressions using this same hierarchy.