AQA GCSE Mathematics Paper 1 (Higher), June 2025: Question 20

3 marks · Medium difficulty · Multi-step Problem

Express the sum of square roots $\sqrt{44} + \sqrt{99} + \sqrt{275}$ in the simplified surd form $a\sqrt{n}$.

Practise this question

Question

Question 20 reads: 'Express sqrt(44) + sqrt(99) + sqrt(275) in the form a sqrt(n) where a and n are integers. [3 marks]'. Below the question statement are blank lines for working and an answer line at the bottom.
Question text

20 Express 44 + 99 + 275 in the form a n where a and n are integers.

[3 marks]

Answer

Mark scheme

Show the mark scheme Mark scheme table for Question 20 shows: M2 for '2 sqrt(11) and 3 sqrt(11) and 5 sqrt(11)' with a note 'M1 any one or two correct values'; and A1 for the final answer '10 sqrt(11)' with comment 'condone a = 10 and n = 11'.

2 11 and 3 11 and 5 11 M2 M1 any one or two correct values

10 11 A1 condone a = 10 and n = 11

How to answer it

Simplifying and Adding Surds into the Form a√n

📋 What this question tests

This question assesses your ability to manipulate radical expressions (surds) on a non-calculator paper. Specifically, it tests whether you can:

  • Identify square factors inside surds using the rule √(a × b) = √a × √b.
  • Express individual surds in simplified form with a common radicand.
  • Collect like surds algebraically to reach the single target form a√n .

Question 20 (3 Marks)

Express √44 + √99 + √275 in the form a√n where a and n are integers.

📐 Step-by-Step Calculation

  1. Look for a common square-free factor:
    Notice that 44, 99, and 275 are all multiples of 11:
    44 = 4 × 11
    99 = 9 × 11
    275 = 25 × 11
  2. Simplify each surd individually:
    • √44 = √(4 × 11) = √4 × √11 = 2√11
    • √99 = √(9 × 11) = √9 × √11 = 3√11
    • √275 = √(25 × 11) = √25 × √11 = 5√11
  3. Collect like surds:
    2√11 + 3√11 + 5√11 = (2 + 3 + 5)√11 = 10√11

✅ Correct Answer & Mark Allocation

Final Answer: 10√11

(Examiners will also accept explicit statements like a = 10, n = 11 )

Mark Scheme Breakdown:
  • [M1]: Correctly simplifies any one or two surds (e.g. finds 2√11 or 3√11 ).
  • [M2]: Correctly simplifies all three surds: 2√11 , 3√11 , and 5√11 .
  • [A1]: Fully correct simplified answer: 10√11 .

💡 Key Knowledge

  • The Multiplication Rule: √(ab) = √a × √b. You can pull square numbers out from under the radical symbol.
  • Square Numbers to Spot: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121... Always look for these factors first!
  • Surds as Algebraic Terms: Think of √11 like x . Just as 2x + 3x + 5x = 10x , adding 2√11 + 3√11 + 5√11 equals 10√11 .

🧠 Exam Technique & Examiner Tips

  • Start with the easiest number: 44 is obviously 4 × 11. That immediately gives you the clue that n = 11 for all three terms!
  • Division check for large values: If you aren't sure how to factorise 275, divide it by 11:
    275 ÷ 11 = 25 (a square number!).
  • Show full method: Writing out the simplified form of each individual term guarantees you secure the method marks (M1/M2) even if you make an addition error at the very end.

❌ Common Errors to Avoid

  • Adding under the square root: Adding values directly inside:
    √(44 + 99 + 275) = √418 .
    Rule: √(A) + √(B) ≠ √(A + B)!
  • Adding the radicands: Writing 2√11 + 3√11 + 5√11 = 10√33 . You only add the coefficients in front, never the number under the square root.
  • Arithmetic slips on 275: Guessing factors rather than doing short division by 11.

Topics

Number · 3.1.1 Structure and calculation

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 1 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.