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

$\frac{\sqrt{10}-\sqrt{5}}{2}$

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

Insert a rational number and an irrational number between the following:

$2.357$ and $3.121$

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

Simplify the following:

$3 \sqrt{3}+2 \sqrt{27}+\frac{7}{\sqrt{3}}$

Express $0.5 \overline{7}$ in the form $\frac{P}{q} ;$ where $p$ and $q$ are integers and $q \neq 0$