Surds (AQA GCSE Further Maths): Exam Questions

Rationalise and simplify $\frac{\sqrt{3}-7}{\sqrt{3}+1}$

Give your answer in the form $a+b\sqrt{3}$ where $a$ and $b$ are integers.

Rationalise the denominator and simplify fully    $\frac{21-11\sqrt{5}}{3-\sqrt{5}}$

Solve   $\sqrt{125}+\sqrt{20}=\sqrt{80}+\sqrt{x}$

$x=........................$

Solve

$5x=x\sqrt{2}+1$

giving your answer in the form

$\frac{a+\sqrt{2}}{b}$

where $a$ and $b$ are integers to be found.

$\sqrt{150}+\sqrt{600}+\sqrt{k}=18\sqrt{6}$

Work out the value of $k$.

$ABC$ is a triangle.

The perpendicular from $A$ meets $BC$ at $D$.

$BC=(6+2\sqrt{7})\mathrm{cm}$

Area of triangle $ABC=(13+3\sqrt{7}){\mathrm{cm}}^2$

Work out the length, in cm, of $AD$.

Give your answer in the form $a+b\sqrt{c}$ where $a,b$ and $c$ are integers.

........................cm

Rationalise the denominator and simplify fully $\frac{1+\sqrt{6}}{\sqrt{2}+2\sqrt{3}}$

Give your answer in the form $\frac{\sqrt{p}}{p}$ where $p$ is a prime number which you should find.

You are given that ${\left(2+\sqrt{a}\right)}^3\equiv 80+b\sqrt{3}$ where $a$ and $b$ are positive integers.

Find $a$ and $b$.