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### Answers (1)

(i) Answer.   2.3093
Solution. Given: $\frac{4}{\sqrt{3}}$
Rationalising,
$\frac{4}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}= \frac{4\sqrt{3}}{3}$
(Given that $\sqrt{3}= 1.732$)
$= \frac{4\times 1\cdot 732}{3}$
= 2.3093
Hence the answer is 2.3093

(ii) Answer.  2.449
Solution.  Given: $\frac{6}{\sqrt{6}}$
Rationalising,
$\frac{6}{\sqrt{6}}\times \frac{\sqrt{6}}{\sqrt{6}}= \frac{6\sqrt{6}}{6}$
$= \frac{6\times \sqrt{2}\sqrt{3}}{6}$
Putting the given values,
$\sqrt{2}= 1\cdot 414,\sqrt{3}= 1\cdot 732$
We get :
$= \sqrt{2}\cdot \sqrt{3}$
$= 1\cdot 414\times 1\cdot 732= 2\cdot 449$
Hence the answer is 2.449

(iii) Answer.  0.462852
Solution.   Given that $\frac{\sqrt{10}-\sqrt{5}}{2}$
This can be written as

$\frac{\sqrt{2}\times \sqrt{5}-\sqrt{5}}{2}$
Now putting the given values,

$\sqrt{2}= 1\cdot 414,\sqrt{5}= 2\cdot 236$
We get :
$\Rightarrow \frac{1\cdot 414\times 2\cdot 236-2\cdot 236}{2}$
= 0.462852
Hence the answer is 0.462852

(iv) Answer.  0.414
Solution.  Given: $\frac{\sqrt{2}}{2+\sqrt{2}}$
Rationalising,
$\frac{\sqrt{2}}{2+\sqrt{2}}\times \frac{2-\sqrt{2}}{2-\sqrt{2}}$
Using   (a – b) (a + b) = a2 – b2
$= \frac{\sqrt{2}\left ( 2-\sqrt{2} \right )}{2^{2}-\sqrt{2}^{2}}$
$= \frac{2\sqrt{2}-2}{4-2}$
$= \frac{2\left ( \sqrt{2}-1 \right )}{2}$
$= \sqrt{2}-1$

Putting the given value of $\sqrt{2}= 1\cdot 414$
We get
= 1.414 – 1
= 0.414
Hence the answer is 0.414

(v)  Answer.  0.318
Solution. Given that $\frac{1}{\sqrt{3}+\sqrt{2}}$
Rationalising,
$\frac{1}{\sqrt{3}+\sqrt{2}}\times \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}$
Using   (a – b) (a + b) = a2 – b2
$= \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}^{2}-\sqrt{2}^{2}}$
$= \frac{\sqrt{3}-\sqrt{2}}{3-2}$
$= \sqrt{3}-\sqrt{2}$

Putting the given values, $\sqrt{2}= 1\cdot 414,\sqrt{3}= 1\cdot 732$
We get,
= 1.732 – 1.414
= 0.318
Hence the answer is 0.318

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