A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Surds and Rationalising the Denominator — mark scheme explained
The short answer
Surds are numbers left in their root form, typically involving square roots. They are used to express exact values without approximation. This section covers how to manipulate surds, including simplifying them and rationalising denominators.
The question
Simplify √48.
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
Marks are typically awarded for each step in the simplification process, including factorisation, application of properties, and final simplification. For rationalising the denominator, marks are given for correctly identifying the conjugate and performing the multiplication.
What the command words demand
- Simplify
- Simplify the given surd or expression involving surds.
- Rationalise
- Rationalise the denominator of the given fraction.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Aim to spend about 2-3 minutes on questions involving surds and rationalising the denominator. Ensure you have enough time to check your work and simplify fully.
- Factorise the number under the root: √48 = √(16 × 3).1 mark
- Use the property of square roots that √(a × b) = √a × √b: √48 = √16 × √3.1 mark
- Simplify the perfect square: √16 = 4, so √48 = 4√3.1 mark
Final answer: 4√3
Work through every step correctly and you earn all 3 marks.
Another worked example
Rationalise the denominator of (2 + √5) / (√7 - √3).
- Multiply the numerator and the denominator by the conjugate of the denominator, which is (√7 + √3): [(2 + √5)(√7 + √3)] / [(√7 - √3)(√7 + √3)].2 marks
- Simplify the denominator using the difference of squares: (7 - 3) = 4.1 mark
- Expand the numerator: (2√7 + 2√3 + √5√7 + √5√3).2 marks
- Simplify the expanded form: (2√7 + 2√3 + √35 + √15) / 4.1 mark
Final answer: (2√7 + 2√3 + √35 + √15) / 4
Work through every step correctly and you earn all 6 marks.
Common mistakes
Forgetting to simplify surds fully.
Why it happens: Students often stop simplifying too early, leaving perfect squares under the root sign.
Fix: Always check if the number under the root can be further factorised into a product of a perfect square and another number.
Incorrectly applying the distributive property when multiplying surds.
Why it happens: Students may forget to distribute the multiplication across all terms in the expression.
Fix: Ensure that each term in one set of parentheses is multiplied by each term in the other set of parentheses.
Failing to rationalise the denominator completely.
Why it happens: Students may multiply only part of the numerator or denominator, leaving surds in the denominator.
Fix: Always multiply both the numerator and the denominator by the conjugate of the denominator.
Incorrectly simplifying expressions involving the difference of squares.
Why it happens: Students may forget to apply the formula (a - b)(a + b) = a 2 - b 2 correctly.
Fix: Practice using the difference of squares formula and double-check your work.
Forgetting to simplify the final answer after rationalising the denominator.
Why it happens: Students may leave the answer in a more complex form than necessary.
Fix: Always check if the numerator can be simplified further after rationalising the denominator.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Simplify Surds | Rewrite a surd in its simplest form by extracting the largest square factor. | 3 |
| Rationalise The Denominator | Remove the surd from the denominator by multiplying by its conjugate and simplify. | 6 |
| Multiply Surd Conjugates | Expand and simplify the product of two conjugate surd expressions to a rational number. | 4 |
| Rationalise The Denominator | Remove the surd from the denominator of a fraction by multiplying to simplify fully. | 4 |
| Total across these question types | 17 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.