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
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
Factorise the number under the root: √48 = √(16 × 3).
- S2
Use the property of square roots that √(a × b) = √a × √b: √48 = √16 × √3.
- S3
Simplify the perfect square: √16 = 4, so √48 = 4√3.
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Factorise the number under the root: √48 = √(16 × 3).
- S2
Use the property of square roots that √(a × b) = √a × √b: √48 = √16 × √3.
- S3
Simplify the perfect square: √16 = 4, so √48 = 4√3.
Final answer: 4√3
Common mistakes
- Forgetting to simplify surds fully. — 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. — 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. — Always multiply both the numerator and the denominator by the conjugate of the denominator.
- Incorrectly simplifying expressions involving the difference of squares. — Practice using the difference of squares formula and double-check your work.
- Forgetting to simplify the final answer after rationalising the denominator. — Always check if the numerator can be simplified further after rationalising the denominator.
Where the marks go
- Full worked solution (all marking points)3 marks