Grade 11 Maths · Algebra

Rationalise a denominator

Remove a square root from the bottom of a fraction by multiplying top and bottom by the right surd (for example 1/√2 = √2/2).

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Grade 11 Maths Surds: Rationalizing the denominator of surdsMiss Martins Maths and Science

Rationalizing Denominators with ConjugatesKhan Academy

GCSE Maths - How to Rationalise the Denominator of a Surd (Part 3/3) (2026/27 exams)Cognito

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Question 2.3 · June 2024 · Paper 1 · Provincial · 3 marks

2.3 WITHOUT using a calculator, show that

\frac{9 - 54}{6 2}

is equal to

\frac{3 2 - 2 3}{4}

.(3)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: rationalise a denominator

Work each one out first, then open the memo and check every tick.

3. Question 2.1.2 · March 2025 · Paper 1 · Provincial · 4 marks
2.1 Simplify the following without using a calculator. Your answers must be free of negative exponent and irrational exponents.
2.1.2 $\frac{( 3 - 3 ) ^{2}}{3 \cdot 6}$ (4)
Show the memo (3 ticks)Memo
1. ✓
  Multiplying out the numerator $\frac{( 3 - 3 ) ^{2}}{3 \cdot 6} = \frac{9 - 6 3 + 3}{3 \cdot 6}$
  $= \frac{12 - 6 3}{3 \cdot 6}$
  1 mark · Square a binomial
2. ✓
  Common factor $= \frac{6 ( 2 - 3 )}{3 \cdot 6}$
  1 mark · Take out a common factor
3. ✓
  Answer $= \frac{6 ( 2 - 3 )}{3 \cdot 6} \times \frac{6}{6} = \frac{6 6 ( 2 - 3 )}{6 3}$
  $= 2 (2 - 3)$ or $22 - 6$
  2 marks · Rationalise a denominator
4. Question 2.1.3 · March 2024 · Paper 1 · Provincial · 4 marks
2.1 Simplify fully without using a calculator. Your answer must be free of negative exponents and irrational denominators.
2.1.3 $\frac{( 3 - 3 ) ^{2}}{3 \cdot 6}$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Expand numerator and simplify denominatorNumerator: $9 - 63 + 3 = 12 - 63$
  Denominator: $18 = 32$
  1 mark · Square a binomial
2. ✓
  Factorise and simplify fraction $\frac{6 ( 2 - 3 )}{3 2}$
  $= \frac{2 ( 2 - 3 )}{2}$
  1 mark · Simplify an algebraic fraction
3. ✓
  Rationalise denominator $\frac{2 ( 2 - 3 )}{2} \times \frac{2}{2}$
  $= \frac{2 2 ( 2 - 3 )}{2}$
  1 mark · Rationalise a denominator
4. ✓
  Final answer $22 - 6$
  1 mark · Simplify and combine surds
5. Question 2.1 · March 2020 · Provincial · 3 marks
2.1 Simplify WITHOUT the use of a calculator:
$18 + 22 + \frac{1}{2}$
Write your answer with a rational denominator.(3)
Show the memo (3 ticks)Memo
1. ✓
  Simplify surds $18 = 32$
  Expression becomes: $32 + 22 + \frac{1}{2}$
  1 mark · Simplify and combine surds
2. ✓
  Rationalise denominator $52 + \frac{1}{2} \times \frac{2}{2}$
  $= 52 + \frac{2}{2}$
  1 mark · Rationalise a denominator
3. ✓
  Final answer $\frac{10 2 + 2}{2} = \frac{11 2}{2}$
  1 mark · Simplify and combine surds
6. Question 2.3 · November 2023 · Paper 1 · Provincial · 5 marks
2.3 Given that $x = 3 - 2$ , simplify $\frac{x ^{2} + 1}{x ^{2} - 5}$ without using a calculator.
(Give your answer in simplest surd form.)(5)
Show the memo (5 ticks)Memo
1. ✓
  Substitution $\frac{x ^{2} + 1}{x ^{2} - 5} = \frac{( 3 - 2 ) ^{2} + 1}{( 3 - 2 ) ^{2} - 5}$
  1 mark · Substitute into a formula
2. ✓
  $\frac{8 - 4 3}{2 - 4 3}$ $= \frac{3 - 4 3 + 4 + 1}{3 - 4 3 + 4 - 5}$
  $= \frac{8 - 4 3}{2 - 4 3}$
  1 mark · Square a binomial
3. ✓
  Rationalisation $= \frac{( 8 - 4 3 ) ( 2 + 4 3 )}{( 2 - 4 3 ) ( 2 + 4 3 )}$
  1 mark · Rationalise a denominator
4. ✓
  Simplification $= \frac{16 + 32 3 - 8 3 - 16 \cdot 3}{4 - 16 \cdot 3}$
  $= \frac{24 3 - 32}{- 44}$
  1 mark · Simplify and combine surds
5. ✓
  Answer $= \frac{8 - 6 3}{11}$
  1 mark · Simplify and combine surds

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