Rationalise the denominator of $\frac{5}{\sqrt{3}-\sqrt{5}}$.

Find the value of $a$ in the following:
$\frac{6}{3 \sqrt{2}-2 \sqrt{3}}=3 \sqrt{2}-a \sqrt{3}$

Simplify the following expressions :
$(i)$ $(5+\sqrt{7})(2+\sqrt{5})$
$(ii)$ $(5+\sqrt{5})(5-\sqrt{5})$
$(iii)$ $(\sqrt{3}+\sqrt{7})^{2}$
$(iv)$ $(\sqrt{11}-\sqrt{7})(\sqrt{11}+\sqrt{7})$

Find the value of the following expression correct to three decimal places,rationalizing the denominator if needed. Take $\sqrt{2} = 1.414$,$\sqrt{3} = 1.732$,and $\sqrt{5} = 2.236$.
$\frac{\sqrt{2}}{2+\sqrt{2}}$

Add $4 \sqrt{6}-8 \sqrt{10}$ and $3 \sqrt{6}+12 \sqrt{10}$.

Rationalise the denominator of the following expression and evaluate it by taking $\sqrt{2}=1.414, \sqrt{3}=1.732$,and $\sqrt{5}=2.236$ up to three decimal places:
$\frac{\sqrt{2}}{2+\sqrt{2}}$