GCSE · Surds

Rationalising the denominatorExam-style questions

Why multiplying top and bottom by the same thing changes nothing, how √a in a denominator is cleared by √a, and how a conjugate clears a two-term denominator like 3 + √5.

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16 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
Write $\dfrac{2 + \sqrt{3}}{2 - \sqrt{3}}$ in the form $a + b\sqrt{3}$ , where $a$ and $b$ are integers.
Level 5 · 3 marksTry it
2
Write $\dfrac{1}{\sqrt{5} - 2} - \sqrt{5}$ in its simplest form.
Level 5 · 3 marksTry it
3
Write $\dfrac{5 - \sqrt{3}}{1 + \sqrt{3}}$ in the form $a + b\sqrt{3}$ , where $a$ and $b$ are integers.
Level 5 · 3 marksTry it
4
$\dfrac{4}{\sqrt{2}} + \sqrt{8} = k\sqrt{2}$ , where $k$ is an integer.

Work out the value of $k$ .
Level 5 · 3 marksTry it
5
Write $\dfrac{11}{5 - \sqrt{3}} - \dfrac{2}{\sqrt{3}}$ in the form $\dfrac{a + b\sqrt{3}}{c}$ where $a$ , $b$ and $c$ are integers.

(4 marks)
Level 5 · 4 marksTry it

2 more questions on rationalising the denominator

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