Order of Operations (BIDMAS/BODMAS) — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers the order of operations: the rules that decide which part of a calculation to do first. By the end of this guide you should be able to say what each letter of BIDMAS stands for, work confidently through brackets including nested ones, deal with indices and roots, and handle the two pairs of equal-priority operations — division with multiplication, and addition with subtraction — in the correct left-to-right order.

You should also be able to apply the rules to longer calculations containing several operations at once, to fraction bars and square roots that act as invisible brackets, and to negative numbers. Finally, you should be able to spot where a calculator will and will not apply the rules for you.

This is one of the few topics on the specification that affects every other topic. A perfectly correct method in algebra, trigonometry or statistics still produces a wrong answer if the arithmetic inside it is evaluated in the wrong order, so the time spent here pays back across the whole course.

Key terms and definitions

BIDMAS / BODMAS — the memory aid for the order of operations: Brackets, Indices (or Orders), Division and Multiplication, Addition and Subtraction. The two versions mean exactly the same thing.

Operation — an arithmetic action: addition, subtraction, multiplication, division, raising to a power or taking a root.

Brackets — grouping symbols showing that whatever is inside must be evaluated as a single quantity before anything else is done with it.

Indices — powers and roots. The index is the small raised number, so in 5³ the index is 3.

Term — a part of an expression separated from the rest by a + or − sign. In 3 × 4 + 7, the terms are 3 × 4 and 7.

Equal priority — two operations ranked the same, so neither automatically comes first; they are worked from left to right in the order they appear.

Core concepts

Why an order is needed at all

Consider 2 + 3 × 4. Working strictly left to right gives 5 × 4 = 20. Doing the multiplication first gives 2 + 12 = 14. Both look reasonable, and that is precisely the problem: without an agreed convention, the same written calculation would have two different answers.

Mathematics therefore fixes one order, and everyone — every textbook, every examiner, every calculator — follows it. The correct answer above is 14, because multiplication outranks addition. BIDMAS is not a rule about how numbers behave; it is an agreement about how to read what has been written.

Brackets come first

Anything inside brackets is worked out before it interacts with anything outside them. In 2 × (3 + 4), the bracket forces the addition first: 3 + 4 = 7, then 2 × 7 = 14. Without the bracket, 2 × 3 + 4 would give 6 + 4 = 10.

Where brackets are nested inside other brackets, work from the innermost outwards. In 3 × (2 + (5 − 1)), evaluate 5 − 1 = 4 first, then 2 + 4 = 6, then 3 × 6 = 18.

Two things behave as brackets even though none are written. A fraction bar groups everything above it and everything below it, so in (7 + 5) ÷ (3 + 1) the brackets are often written as a fraction with 7 + 5 on top and 3 + 1 underneath; either way the answer is 12 ÷ 4 = 3. A square root sign groups everything underneath it in the same way: √(9 + 16) means √25 = 5, not 3 + 4.

Indices come second

After brackets, deal with powers and roots. In 5 + 2³, the index is evaluated first: 2³ = 8, so the answer is 13. In 3 × 4², the square applies only to the 4, giving 3 × 16 = 48 — not (3 × 4)² = 144.

This is worth dwelling on, because it is a common source of error. An index attaches to the number immediately before it and to nothing else, unless brackets say otherwise. So 2 × 3² is 2 × 9 = 18, while (2 × 3)² is 6² = 36.

Division and multiplication share third place

Division and multiplication have equal priority. Neither outranks the other, so they are worked from left to right in the order written.

In 12 ÷ 2 × 3, the division comes first because it is further left: 12 ÷ 2 = 6, then 6 × 3 = 18. Doing the multiplication first would give 12 ÷ 6 = 2, which is wrong.

The letters D and M appearing in that order in BIDMAS is an accident of the mnemonic, not a statement that division outranks multiplication. Reading it as a strict ranking is one of the most frequent mistakes in this topic.

Addition and subtraction share fourth place

The same applies at the bottom of the order. Addition and subtraction have equal priority and are worked from left to right.

In 10 − 4 + 2, the subtraction is further left, so: 10 − 4 = 6, then 6 + 2 = 8. Treating the A of BIDMAS as outranking the S would give 10 − 6 = 4, which is wrong.

Negative numbers

Negative signs follow the same rules, but two points cause trouble. First, a minus sign directly in front of a bracket multiplies everything inside it: 10 − (3 + 2) = 10 − 5 = 5.

Second, squaring a negative number requires brackets to make the intention clear. (−3)² means (−3) × (−3) = 9, whereas −3² is read as −(3²) = −9. The two differ, and an exam question that writes one and expects the other is testing exactly this.

Calculators

A scientific calculator applies BIDMAS automatically, which is helpful but can hide errors. It will faithfully evaluate whatever you type, so a missing bracket produces a confidently wrong answer.

Typing 12 ÷ 2 + 4 gives 10, but if the intended calculation was 12 divided by the whole of 2 + 4, you must type 12 ÷ (2 + 4) to get 2. When a fraction bar appears in a question, put brackets around the entire numerator and the entire denominator before dividing.

Worked examples

Example 1: A calculation with three operation types

Work out 4 + 3 × (8 − 5)².

Take the layers in order. The bracket first: 8 − 5 = 3, giving 4 + 3 × 3².

Then the index. It applies to the 3 that came from the bracket, so 3² = 9, giving 4 + 3 × 9.

Then multiplication, which outranks addition: 3 × 9 = 27, giving 4 + 27.

Finally the addition: 31.

Notice that each step removed exactly one layer, and the expression was rewritten every time. That habit prevents the commonest error of all, which is losing track of what has already been done.

Example 2: Equal priority worked left to right

Work out 20 ÷ 4 × 2 + 6 − 1.

There are no brackets and no indices, so start with the division and multiplication group, taken left to right. 20 ÷ 4 = 5, then 5 × 2 = 10. The expression is now 10 + 6 − 1.

Now the addition and subtraction group, also left to right: 10 + 6 = 16, then 16 − 1 = 15.

A student who treated D as outranking M would compute 4 × 2 = 8 first, then 20 ÷ 8 = 2.5, and reach 7.5 — a completely different answer from one misreading of the mnemonic.

Example 3: An invisible bracket

Work out (5 + 7) ÷ (9 − 3), and then √(16 + 9).

For the first, both brackets must be cleared before dividing: 5 + 7 = 12 and 9 − 3 = 6, so 12 ÷ 6 = 2.

For the second, the root sign groups everything beneath it: 16 + 9 = 25 first, then √25 = 5. Taking the roots separately would give 4 + 3 = 7, which is wrong. This is the single most common error involving roots.

Common mistakes and how to avoid them

Reading BIDMAS as six strict ranks. It is four ranks, because division and multiplication share one and addition and subtraction share another. Within each shared rank, work left to right.

Applying an index to too much. In 3 × 4², the square belongs to the 4 alone. If the whole product is meant to be squared, brackets must say so.

Losing the invisible brackets. A fraction bar and a square root sign both group everything they cover. Write those brackets in explicitly before reaching for a calculator.

Mishandling −3² and (−3)². The first is −9 and the second is +9. Check which one the question has written.

Doing everything at once. Rewriting the full expression after each step takes seconds and eliminates most errors. Trying to hold three operations in your head does not.

Trusting the calculator to know what you meant. It applies the rules to what you typed, not to what was printed in the question.

Exam technique for "Order of Operations"

Underline or circle the brackets before starting. It takes a moment and makes the structure of the calculation visible.

Write each stage on a new line. Marks in multi-step questions are frequently awarded for method, so a visible sequence of steps can earn credit even when the final answer is wrong. A single line with one number on it cannot.

When a question gives an answer and asks you to insert brackets to make it correct, work backwards: decide which operation must have happened first to produce that answer, then place the brackets to force it.

If a calculation involves a fraction, evaluate the top and the bottom completely before dividing. Most calculator errors in this topic come from dividing too early.

Check the answer for plausibility. If 2 + 3 × 4 came out as 20, the multiplication was done second, and the error is immediately identifiable.

Quick revision summary

BIDMAS gives four ranks, not six: Brackets, then Indices, then Division and Multiplication together, then Addition and Subtraction together.

Within each of the two shared ranks, work strictly left to right. 12 ÷ 2 × 3 = 18, and 10 − 4 + 2 = 8.

Nested brackets are worked from the inside out. Fraction bars and square root signs act as brackets even though none are printed.

An index attaches only to the quantity immediately before it: 3 × 4² = 48, but (3 × 4)² = 144. And −3² = −9 while (−3)² = +9.

Rewrite the whole expression after every step. It costs a few seconds and prevents most of the marks lost in this topic.