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

1. Why Order of Operations Matters

Consider the expression $3 + 4 \times 2$. If you work left to right you get $7 \times 2 = 14$. If you multiply first you get $3 + 8 = 11$. Both cannot be right — and the agreed answer is $11$.

Mathematics needs one universal rule so that every person — and every calculator — reaches the same answer. That rule is captured by the acronym BIDMAS (sometimes written BODMAS): Brackets, Indices (Orders), Division, Multiplication, Addition, Subtraction.

Notice that Division and Multiplication share the same priority level — you do not always divide before you multiply. You work through them left to right as they appear. The same applies to Addition and Subtraction.

2. Brackets and Why They Come First

Brackets are a deliberate instruction to the reader: deal with this part first. They override every other rule in BIDMAS, which is exactly why they are so useful for controlling the order of a calculation.

Worked example: Evaluating an Expression with Brackets

Calculate $5 \times (3 + 4) - 2$

1. $(3 + 4)$ — Brackets come first in BIDMAS. Evaluate the contents before anything else. [M1]
2. $3 + 4 = 7$ — The expression now reads $5 \times 7 - 2$.
3. $5 \times 7 = 35$ — Multiplication comes before subtraction in BIDMAS. Expression reads $35 - 2$. [M1]
4. $35 - 2 = 33$ — Final step — addition and subtraction last. [A1]

Answer: $33$

Without the brackets, $5 \times 3 + 4 - 2 = 15 + 4 - 2 = 17$ — a completely different result. Brackets change the answer, so always spot them first.

3. Indices — Powers and Roots

After brackets come indices. This level covers both powers (such as $4^3$) and roots (such as $\sqrt{25}$ or $\sqrt[3]{8}$). Roots are the inverse of powers, so they sit at the same priority level.

A common slip is to multiply before squaring. For example, in $3 \times 2^2$, you must square the $2$ first: $2^2 = 2 \times 2 = 4$, giving $3 \times 4 = 12$. Working left to right would incorrectly give $(3 \times 2)^2 = 36$.

Worked example: Expression with a Power and a Root

Calculate $\sqrt{36} + 2^3 - 5$

1. No brackets present. — Move to the next level: indices (powers and roots).
2. $\sqrt{36} = 6$ — Square root is an index-level operation. $6 \times 6 = 36$, so $\sqrt{36} = 6$. [M1]
3. $2^3 = 2 \times 2 \times 2 = 8$ — Cube means multiply by itself three times. Expression now reads $6 + 8 - 5$. [M1]
4. $6 + 8 = 14$, then $14 - 5 = 9$ — Addition and subtraction share equal priority; work left to right. [A1]

Answer: $9$

4. Division, Multiplication, and Reciprocals

Division and multiplication sit at the same level in BIDMAS. When both appear in an expression, you work through them left to right — not always dividing before multiplying.

Reciprocals connect directly to division. The reciprocal of $n$ is $\frac{1}{n}$, and dividing by $n$ is identical to multiplying by $\frac{1}{n}$. This equivalence is especially useful when $n$ is itself a fraction.

Worked example: Left-to-Right Rule for Division and Multiplication

Calculate $24 \div 4 \times 3$

1. $24 \div 4 \times 3$ — Division and multiplication share equal priority, so work left to right.
2. $24 \div 4 = 6$ — Expression now reads $6 \times 3$. [M1]
3. $6 \times 3 = 18$ — Final answer. [A1]

Answer: $18$

Had you multiplied first — $4 \times 3 = 12$, then $24 \div 12 = 2$ — you would get a different, incorrect answer. The left-to-right rule prevents this.

Worked example: Using a Reciprocal in a Calculation

Calculate $15 \div \frac{3}{5}$

1. Reciprocal of $\frac{3}{5}$ is $\frac{5}{3}$ — Dividing by a fraction is the same as multiplying by its reciprocal (flip the fraction). [M1]
2. $15 \times \frac{5}{3}$ — The expression is now a straightforward multiplication.
3. $15 \times \frac{5}{3} = \frac{15 \times 5}{3} = \frac{75}{3} = 25$ — $15 \times 5 = 75$, then $75 \div 3 = 25$. Notice the result is larger than $15$ — dividing by a number less than $1$ always increases the value. [A1]

Answer: $25$

5. Putting It All Together — Multi-Step Expressions

Exam questions often combine all levels of BIDMAS in a single expression. The method is always the same: scan the whole expression, then work through the priority levels one at a time — B, then I, then D/M left to right, then A/S left to right.

Worked example: Full BIDMAS Evaluation

Calculate $(10 - 4)^2 \div 4 + \sqrt{9} \times 3$

1. $(10 - 4) = 6$ — Brackets first. Expression becomes $6^2 \div 4 + \sqrt{9} \times 3$. [M1]
2. $6^2 = 6 \times 6 = 36$ — Indices next. Expression becomes $36 \div 4 + \sqrt{9} \times 3$.
3. $\sqrt{9} = 3$ — Still at the indices level. Expression becomes $36 \div 4 + 3 \times 3$. [M1]
4. $36 \div 4 = 9$, then $3 \times 3 = 9$ — Both operations are at the same level; handle left to right. Expression becomes $9 + 9$.
5. $9 + 9 = 18$ — Addition is the final step. [A1]

Answer: $18$

6. Exam Technique and Common Mistakes

Most marks lost on BIDMAS questions come from one of three predictable errors. Knowing them in advance means you can actively check for them before writing your final answer.

Mastering BIDMAS means you can evaluate any expression confidently — whether it contains brackets, powers, roots, reciprocals, or all four at once. Every rule in this lesson traces back to the single specification requirement: use conventional notation for the priority of operations.

One final exam habit makes BIDMAS reliable under pressure: rewrite the expression in layers rather than trying to hold every step in your head. Evaluate brackets first, substitute those results back into the original expression, then work through indices, division and multiplication from left to right before finishing with addition and subtraction. If two operations share the same priority, never jump to the one that looks easier. A quick line-by-line rewrite prevents the classic errors of adding too early, ignoring an index, or forgetting that division and multiplication are equal-priority operations worked in order.