Surds (Cambridge (CIE) IGCSE International Maths: Extended): Exam Questions

Simplify.

   $\sqrt{32}+\sqrt{98}$

Rationalise the denominator of $\frac{12}{\sqrt{3}}$.

Work out the value of  ${\left(\sqrt{2}+\sqrt{8}\right)}^2$.

Work out the value of        ${\left(\sqrt{12}-\sqrt{3}\right)}^2$.

Rationalise the denominator of    $\frac{10}{\sqrt{5}}$. Give your answer in its simplest form.

Rationalise the denominator of $\frac{10}{\sqrt{6}}$

Give your answer in its simplest form

Show that $\sqrt{20}=2\sqrt{5}$.

Show that $\frac{\sqrt{12}}{\sqrt{3+2}}$

can be written in the form $a\sqrt{b}$where $a$ is a simplified fraction and $b$ is an integer.

Write  ${\left(1+\sqrt{3}\right)}^2$   in the form  $a+b\sqrt{3}$.

Simplify $\sqrt{6}\times \sqrt{3}$.

Show that 

Show each stage of your working.

Simplify fully by rationalising the denominator.

$\frac{20}{\sqrt{5}}$

Show that $\frac{\sqrt{20}+\sqrt{80}}{\sqrt{3}}$ can be expressed in the form $\sqrt{a}$ where $a$ is an integer.

Show your working clearly.

Show that $\sqrt{45}+\sqrt{20}=5\sqrt{5}$

Show your working clearly.

Simplify fully.

            $\sqrt{200}$

Write $\sqrt{12}+\sqrt{75}$ in the form $k\sqrt{3}$.

Simplify fully.

$\sqrt{50}+\sqrt{2}$

Work out       $2\sqrt{10}\times \sqrt{80}\times \sqrt{18}$

Give your answer as an integer.

Rationalise the denominator and simplify $\frac{10}{3\sqrt{5}}$

$8300=100\times 83$

Circle the number that is closest in value to $\sqrt{8300}$

 Show that $\frac{14}{\sqrt{7}}$  can be written in the form $a\sqrt{b}$ where $a$ and $b$ are integers.

You are given that $\sqrt{7}=2.65$ and $\sqrt{70}=8.37$, each correct to 2 decimal places.

Use this information to find the value of

(i) $\sqrt{700}$

[1]

(ii) $\sqrt{280}$

[1]

Rationalise the denominator.

$\frac{1}{\sqrt{2}+1}$

Rationalise the denominator of $\frac{5}{\sqrt{2}}$.

Expand and simplify ${\left(2+\sqrt{3}\right)}^2-{\left(2-\sqrt{3}\right)}^2$.

Expand   $\left(1+\sqrt{2}\right)\left(3-\sqrt{2}\right)$
Give your answer in the form $a+b\sqrt{2}$ where $a$ and $b$ are integers.

Simplify fully  $\frac{\left(6-\sqrt{5}\right)\left(6+\sqrt{5}\right)}{\sqrt{31}}$

You must show your working.

$\sqrt{5}\left(\sqrt{8}+\sqrt{18}\right)$    can be written in the form  $a\sqrt{10}$ where $a$ is an integer.

Find the value of $a$.

Show that    $\frac{\left(4-\sqrt{3}\right)\left(4+\sqrt{3}\right)}{\sqrt{13}}$ simplifies to  $\sqrt{13}$.

Martin did this question.

Here is how he answered the question.

$\begin{array}{rcl}\frac{14}{2+\sqrt{3}}& =& \frac{14\times \left(2-\sqrt{3}\right)}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}\\ & & \\ & =& \frac{28-14\sqrt{3}}{4+2\sqrt{3}-2\sqrt{3}+3}\\ & =& \frac{28-14\sqrt{3}}{7}\\ & =& 4-2\sqrt{3}\end{array}$

Martin's answer is wrong.

Find Martin's mistake.

Sian did this question.

Here is how she answered the question.

                     $\begin{array}{rcl}\frac{5}{\sqrt{12}}& =& \frac{5\sqrt{12}}{\sqrt{12}\times \sqrt{12}}\\ & & \\ & =& \frac{5\times 3\sqrt{2}}{12}\\ & =& \frac{5\sqrt{2}}{4}\end{array}$

Sian's answer is wrong.

Find Sian's mistake.

Rationalise the denominator of  $\frac{4}{7-\sqrt{5}}$

Show each stage of your working.

Give your answer in the form $a+b\sqrt{5}$ where $a$ and $b$ are fractions in their simplest forms.

Express  $\sqrt{3}+\sqrt{12}$ in the form $a\sqrt{3}$ where $a$ is an integer.

Express ${\left(\frac{1}{\sqrt{3}}\right)}^7$ in the form $\frac{\sqrt{b}}{c}$ where $b$ and $c$ are integers.

Given that $y$ is a prime number,

express $\frac{3}{2-\sqrt{y}}$ in the form $\frac{a+b\sqrt{y}}{c-y}$ where $a$ , $b$ and $c$are integers.

Show that $\frac{14}{3-\sqrt{2}}$ can be written in the form $a+b\sqrt{2}$.

Without using a calculator, rationalise the denominator of $\frac{6}{3-\sqrt{7}}$

Simplify your answer.

You must show each stage of your working.

Show that $\frac{\sqrt{150}-\sqrt{6}}{\sqrt{2}\times \sqrt{3}}$ simplifies to an integer.

Show that  $\frac{\sqrt{8}}{\sqrt{8}-2}$ can be written in the form $n+\sqrt{n}$ , where $n$ is an integer.

Show your working clearly.

Express $\frac{2}{\sqrt{3}-1}$ in the form $p+\sqrt{q}$

where $p$ and $q$ are integers. Show your working clearly.

Show that  ${\left(5\sqrt{3}-\sqrt{12}\right)}^2$  simplifies to an integer.

$$\begin{gathered} a=\sqrt{8}+2 \\ b=\sqrt{8}-2 \\ T=a^2-b^2 \end{gathered}$$

Work out the value of $T$.

Give your answer in the form $c\sqrt{2}$ where $c$ is an integer.

 Show that  $\frac{6-\sqrt{8}}{\sqrt{2}-1}$    can be written in the form $a+b\sqrt{2}$  where $a$ and $b$ are integers.

Show that  $\frac{1}{1+{\displaystyle \frac{1}{\sqrt{2}}}}$    can be written as  $2-\sqrt{2}$

$\begin{array}{rcl}a& =& \sqrt{8}+4\\ b& =& \sqrt{8}-4\end{array}$

$(a–b)(a+b)$ can be written in the form $y\sqrt{4y}$

Find the value of $y$
Show your working clearly.

Rationalise the denominator of $\frac{a+\sqrt{4b}}{a-\sqrt{4b}}$ where $a$ is an integer and $b$ is a prime number.

Simplify your answer.

Show that $\frac{2}{6-3\sqrt{2}}$ can be written in the form $\frac{a+\sqrt{a}}{b}$

where $a$ and $b$ are integers.
Show your working clearly.

Express $\frac{8}{\sqrt{5}-1}$ in the form $\sqrt{a}+b$ where a and b are integers.

Show each stage of your working clearly.

The area of a rectangle is $18$ ${\mathrm{cm}}^2$

The length of the rectangle is $\left(\sqrt{7}+1\right)$ $\mathrm{cm}$.

Without using a calculator and showing each stage of your working, find the width of the rectangle.

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

.....................$\mathrm{cm}$

Show that $\frac{4+\sqrt{8}}{\sqrt{2}-1}$can be written in the form $a+b\sqrt{2}$, where $a$ and $b$ are integers.

Show each stage of your working clearly and give the value of a and the value of $b$.

In the following equation, $n$ is an integer greater than 1.

${\left(\sqrt{2}\right)}^n=k\sqrt{2}$

i) Find $k$ when $n$ = 7.

$k$ = ...................... [2]

ii) Find $n$ when $k$ = 64.

$n$ = ..................... [2]

 Show that $\frac{\left(4+2\sqrt{5}\right)}{\sqrt{5}-1}$ can be simplified to $\frac{3\sqrt{5}+7}{2}$

Here is a sequence.

      5   $5\sqrt{3}$     15   $15\sqrt{3}$

Work out the next term.

Find the $n$th term.

Simplify $\sqrt{80}+\sqrt{2\frac{2}{9}}$

Give your answer in the form  $\frac{a\sqrt{5}}{b}$    where $a$ and $b$ are integers.

Work out    $\sqrt{18}-\frac{28}{\sqrt{50}}$

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

Show that $\frac{2\sqrt{6}}{\sqrt{5}}-\frac{\sqrt{3}}{\sqrt{10}}$ can be written in the form $\frac{c\sqrt{d}}{10}$

where $c$and $d$ are integers.