Simplify the following expressions :

$(i)$ $(5+\sqrt{7})(2+\sqrt{5})$

$(ii)$ $(5+\sqrt{5})(5-\sqrt{5})$

$(iii)$ $(\sqrt{3}+\sqrt{7})^{2}$

$(iv)$ $(\sqrt{11}-\sqrt{7})(\sqrt{11}+\sqrt{7})$

Rationalise the denominator of $\frac{1}{2+\sqrt{3}}$.

Are the square roots of all positive integers irrational ? If not, give an example of the square root of a number that is a rational number.

Represent $ \sqrt{9.3}$ on the number line.

Classify the following numbers as rational or irrational :

$(i)$ $2-\sqrt{5}$

$(ii)$ $(3+\sqrt{23})-\sqrt{23}$

$(iii)$ $\frac{2 \sqrt{7}}{7 \sqrt{7}}$

$(iv)$ $\frac{1}{\sqrt{2}}$

$(v)$ $2 \pi$

Find six rational numbers between $3$ and $4$.