Edexcel IGCSE Math A 1.4.HB Manipulate surds

Practise manipulating surds by expanding brackets, simplifying radicals and rationalising denominators without a calculator.

Exam points

rationalise denominators using a conjugate and simplify into forms
expand and collect surd terms to prove identities or solve equations involving √2 or √3
use surd manipulation in contexts

[Maximum number: 3]

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.

E.g.
$\begin{aligned} \frac{2}{6-3 \sqrt{2}} \times \frac{6+3 \sqrt{2}}{6+3 \sqrt{2}} \text { or } \frac{2}{6-3 \sqrt{2}} \times \frac{-6-3 \sqrt{2}}{-6-3 \sqrt{2}} \end{aligned}$

M1 for rationalising the denominator by
multiplying numerator and denominator
by $6+3 \sqrt{2}$ (or $-6-3 \sqrt{2}$ )

$\begin{aligned} \frac{12+6 \sqrt{2}}{36-18 \sqrt{2}+18 \sqrt{2}-18} \text { or } \frac{12+6 \sqrt{2}}{18} \text { or } \frac{12+6 \sqrt{2}}{6^{2}-(3 \sqrt{2})^{2}} \text { or } \frac{12+6 \sqrt{2}}{6^{2}-9 \times 2} \end{aligned}$

M1 (numerator may be expanded or
denominator may be 4 terms which need
to be all correct)

$\frac{2+\sqrt{2}}{3}$

A1 or for stating a=2 and b=3
for $\frac{2+\sqrt{2}}{3}$ from correct working
dep on M2

All question bank results loaded

1.4.HB Manipulating and rationalising surds question 1