CAIE IGCSE Mathematics Extended E1.18 Surds Questions

Practise simplifying surds, expanding brackets with square roots and rationalising denominators into exact simplified forms.

Exam points

simplify sums or differences of surds by extracting square factors before combining terms
expand brackets with surds and collect terms into forms
rationalise denominators using a single surd factor or a conjugate, then simplify

[Maximum number: 4]

(a)

$(3-\sqrt5)(2+3\sqrt5)=a+b\sqrt5$
Find the value of a and the value of b.
$\begin{aligned}a&=\\b&=\end{aligned}$

$\begin{aligned} & a= \\ & b= \end{aligned}$

[ 2 ]

$\begin{aligned}a&=-9\\b&=7\end{aligned}$

B1 for each
or for three correct terms from $6+9\sqrt5-2\sqrt5-3\sqrt5\sqrt5$ oe

(b)

Rationalise the denominator.

Write your answer in its simplest form.

$\frac{6}{\sqrt{2}}$

[ 2 ]

$3\sqrt2$ final answer

M1 for $\times\frac{\sqrt2}{\sqrt2}$ oe

[Maximum number: 4]

(a)

$\frac{9}{\sqrt{3}}$
Rationalise the denominator.
Give your answer in its simplest form.

[ 2 ]

$3\sqrt3$ cao

M1 for $\frac{9}{\sqrt3}\times\frac{\sqrt3}{\sqrt3}$ oe

(b)

$(5-\sqrt{2})(1+3 \sqrt{2})=c+k \sqrt{2}$

Find the value of c and the value of k.

\begin{aligned} & c= \\ & k= \end{aligned}

[ 2 ]

$\begin{aligned}c&=-1\\ k&=14\end{aligned}$

B1 for each;
or for 3 correct terms from $5+15\sqrt2-\sqrt2-3\sqrt2\sqrt2$ oe

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