5. Indices and Surds
iGCSE (2021 Edition)
5.11 Rationalising the denominator (Extended)
Worksheet
1
Find the conjugate of each of the following:
a
5 + \sqrt{5}
b
6 - \sqrt{v}
c
\sqrt{6} + \sqrt{3}
d
\sqrt{n} - \sqrt{5}
e
\sqrt{5} + y
f
2 - 9 \sqrt{3}
g
3+ 6\sqrt{8}
h
7- 2\sqrt{9}
i
3\sqrt{7} + 4\sqrt{3}
j
9 \sqrt{2} - 3\sqrt{r}
k
4 \sqrt{s} + 8\sqrt{t}
l
2 \sqrt{w} - 9\sqrt{x}
2
Factorise the expression: x^{2} - y^{2}
3
Evaluate the following:
a
\left(\sqrt{3} + \sqrt{10}\right) \times \left(\sqrt{3} - \sqrt{10}\right)
b
\left(\sqrt{3} + \sqrt{10}\right) \times \left(\sqrt{3} + \sqrt{10}\right)
c
\left( - \left( \sqrt{3} + 5 \right)\right) \times \sqrt{3}
d
\left(\sqrt{5} + \sqrt{12}\right) \times \left(\sqrt{5} - \sqrt{12}\right)
e
\left(-2\sqrt{5} - \sqrt{12}\right) \times \sqrt{12}
f
\left(2\sqrt{5} - \sqrt{12}\right) \times \left(5\sqrt{5} - \sqrt{12}\right)
4
Rationalise the denominator of the given expressions. Express your answer in simplest surd form:
a
\dfrac{1}{\sqrt{7}}
b
\dfrac{2}{\sqrt{6}}
c
\dfrac{3}{\sqrt{13}}
d
\dfrac{\sqrt{13}}{\sqrt{2}}
e
\dfrac{\sqrt{21}}{\sqrt{7}}
f
\dfrac{\sqrt{5}}{\sqrt{30}}
g
\dfrac{4 \sqrt{30}}{\sqrt{6}}
h
\dfrac{11 \sqrt{7}}{13 \sqrt{3}}
i
- \dfrac{6 \sqrt{22}}{5 \sqrt{11}}
j
- \dfrac{14 \sqrt{10}}{6 \sqrt{5}}
k
\dfrac{8 \sqrt{32}}{20 \sqrt{7}}
l
- \dfrac{15 \sqrt{18}}{19 \sqrt{8}}
5
Rationalise the denominator of the given expressions. Express your answer in simplest surd form:
a
\dfrac{\sqrt{5} + 9}{\sqrt{7}}
b
\dfrac{\sqrt{5} + 3}{\sqrt{5}}
c
\dfrac{4 - \sqrt{12}}{\sqrt{10}}
d
\dfrac{\sqrt{7} - 3}{\sqrt{3}}
e
\dfrac{10 \sqrt{2} + 7}{\sqrt{11}}
f
\dfrac{15 - 2\sqrt{3}}{\sqrt{2}}
g
\dfrac{3 \sqrt{5} + 12}{\sqrt{20}}
h
\dfrac{6 \sqrt{14} - 11}{\sqrt{5}}
i
\dfrac{\sqrt{39} + \sqrt{6}}{\sqrt{3}}
j
\dfrac{\sqrt{7} - \sqrt{13}}{\sqrt{15}}
k
\dfrac{6 \sqrt{2} + 10 \sqrt{10}}{\sqrt{12}}
l
\dfrac{-20 \sqrt{5} + 7 \sqrt{11}}{\sqrt{6}}
6
Rationalise the denominator of each fraction and then find the sum. Express your answer in simplest surd form:
a
\dfrac{1}{\sqrt{3}} + \dfrac{3}{\sqrt{3}}
b
\dfrac{2}{\sqrt{7}} + \dfrac{1}{\sqrt{7}}
c
\dfrac{1}{\sqrt{2}}+ \dfrac{1}{\sqrt{3}}
d
\dfrac{1}{\sqrt{5}}- \dfrac{1}{\sqrt{10}}
e
\dfrac{2}{\sqrt{14}} + \dfrac{4}{\sqrt{7}}
f
\dfrac{6}{\sqrt{3}}- \dfrac{3}{\sqrt{6}}
g
\dfrac{3}{\sqrt{11}} - \dfrac{5}{\sqrt{22}}
h
\dfrac{\sqrt{2}}{\sqrt{7}} + \dfrac{\sqrt{7}}{\sqrt{2}}
i
\dfrac{\sqrt{10}}{\sqrt{20}} - \dfrac{\sqrt{30}}{\sqrt{40}}
j
\dfrac{2 \sqrt{5}}{\sqrt{5}} + \dfrac{\sqrt{3}}{\sqrt{15}}
k
\dfrac{4}{\sqrt{6}} - \dfrac{1}{2\sqrt{3}}
l
\dfrac{3\sqrt{2}}{4\sqrt{12}} + \dfrac{2\sqrt{10}}{3\sqrt{2}}
7
What is the lowest term that the numerator and denominator of \dfrac{10 \sqrt{13}}{11 \sqrt{11}} must be multiplied by to rationalise the denominator?
8
Rationalise the denominator for the given expressions. Express your answer in simplest surd form:
a
\dfrac{5}{\sqrt{7} - 3}
b
\dfrac{3}{5 \sqrt{2} - 4}
c
\dfrac{5}{9 + \sqrt{3}}
d
\dfrac{27}{\sqrt{11} - \sqrt{2}}
e
\dfrac{4}{\sqrt{6} - \sqrt{7}}
f
\dfrac{3}{4 \sqrt{7} + 8 \sqrt{2}}
g
\dfrac{10}{5 \sqrt{10} - 5\sqrt{3}}
h
\dfrac{2}{7 \sqrt{3} + 2 \sqrt{6}}
i
\dfrac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}
j
\dfrac{9 \sqrt{2} + 3 \sqrt{11}}{9 \sqrt{2} - 3 \sqrt{11}}
k
\dfrac{\sqrt{6} - 3 \sqrt{5}}{\sqrt{6} + 3 \sqrt{5}}
l
\dfrac{3 \sqrt{4} - 4 \sqrt{3}}{4 \sqrt{4} + 2 \sqrt{3}}
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