Surds (Edexcel IGCSE Maths A (Modular): Higher Unit 1): Exam Questions

Show that $\frac{5}{\sqrt{2}}$ can be written as $\frac{5\sqrt{2}}{2}$

Show that ${\left(2+\sqrt{3}\right)}^2-{\left(2-\sqrt{3}\right)}^2=8\sqrt{3}$

Show that $\left(1+\sqrt{2}\right)\left(3-\sqrt{2}\right)=1+2\sqrt{2}$

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.

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

You must show your working.

Show that $\sqrt{5}\left(\sqrt{8}+\sqrt{18}\right)$ can be written in the form $a\sqrt{10}$ and state the value of $a$. You must show your full working.

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

Show that $\sqrt{3}+\sqrt{12}$ can be rewritten as $3\sqrt{3}$

Show that ${\left(\frac{1}{\sqrt{3}}\right)}^7=\frac{\sqrt{3}}{81}$

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

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 $\frac{\sqrt{125}+\sqrt{80}}{\sqrt{3}}$ in the form $\sqrt{n}$ where $n$ is an integer.
Show your working clearly.

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

Show that $T=c\sqrt{2}$ stating the value of $c$

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 $b$.

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

The diagram shows a cuboid with a square cross section.

The volume of the cuboid is $\left(13+6\sqrt{5}\right)$cm³

Without using a calculator, find the value of $x$
Give your answer in the form $a+\sqrt{b}$ where $a$ and $b$ are integers.
Show your working clearly