Surds (Cambridge International IGCSE Mathematics): Exam Questions

Simplify

$\sqrt{32}+\sqrt{98}$

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

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

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

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

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.

Show that ( 6 + 2 sqrt index _ of 12 ) ^2 = 12 ( 7 + 4 sqrt( 3 )

Show each stage of your working.

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.

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

Give your answer in its simplest form

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

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

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

Simplify fully by rationalising the denominator.

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

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 $\frac{10}{3\sqrt{5}}$

$8300=100\times83$

Select 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.

Rationalise the denominator

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

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

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

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

Martin did this question.

Here is how he answered the question.

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

Martin's answer is wrong.

Find Martin's mistake.

Sian did this question.

Here is how she answered the question.

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

Sian's answer is wrong.

Find Sian's mistake.

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

You must show your working.

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

Find the value of $a$.

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

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

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

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.

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.

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 $\frac{\sqrt{8}}{\sqrt{8}-2}$ 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 $\frac{14}{3-\sqrt{2}}$ can be written in the form $a+b\sqrt{2}$.

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

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

Show that $\frac{\sqrt{98}-3\sqrt{2}}{4\sqrt{2}-4}$ can be written as $a+b\sqrt{2}$, where $a$ and $b$ are integers.

Simplify $2\sqrt{12}+3\sqrt{27}-\sqrt{75}$

Rationalise the denominator $\frac{4}{5-\sqrt{3}}$

Give your answer in the form $p+q\sqrt{r}$ where $p$ and $q$ are rational numbers in their simplest form and $r$ is an integer.

$a=\sqrt{8}+2,\,b=\sqrt{8}-2,\,T=a^{2}-b^{2}\,$

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

$\begin{gathered} a = \sqrt{8} + 4 \\ b = \sqrt{8} - 4 \end{gathered}$

$(a--b)(a+b)$ 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 $( \sqrt{7 } + 1 )$ $\mathrm{cm}$.

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.

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

i) Find $k$ when $n$ = 7.

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

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

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

Show that $\frac{( 4 + 2 \sqrt{5} )}{ \sqrt{5} - 1}$ 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.

Extract:

Here is a sequence.

 5   5√3    15   15√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.