Rationalise the denominator of the following expression:
$\frac{4}{\sqrt{10}+\sqrt{6}}$

If $x = 5 + 2\sqrt{6}$,then find the value of $x^{2} + \frac{1}{x^{2}}$ and $x^{3} + \frac{1}{x^{3}}$.

Find three rational numbers between $\frac{1}{7}$ and $\frac{3}{7}$.

Rationalise the denominators of the following:
$(i)$ $\frac{1}{\sqrt{7}}$
$(ii)$ $\frac{1}{\sqrt{7}-\sqrt{6}}$
$(iii)$ $\frac{1}{\sqrt{5}+\sqrt{2}}$
$(iv)$ $\frac{1}{\sqrt{7}-2}$

For each question,select the proper option from the four options given to make the statement true: $(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3}) = \ldots \ldots \ldots$

Rationalise the denominator in each of the following and hence evaluate by taking $\sqrt{2}=1.414, \sqrt{3}=1.732$ and $\sqrt{5}=2.236,$ up to three decimal places.
$\frac{\sqrt{10}-\sqrt{5}}{2}$