Write the following numbers in ascending order:

$\begin{matrix} \text{(i)} & 3\sqrt{2} , 2\sqrt{3} , \sqrt{15} , 4 \\[1.5em] \text{(ii)} & 3\sqrt{2} , 2\sqrt{8} , 4, \sqrt{50} ,4\sqrt{3} \\[1.5em] \end{matrix}$

Write the following real numbers in descending order:

$\begin{matrix} \text{(i)} & \dfrac{9}{\sqrt{2}} , {\dfrac{3}{2}}\sqrt{5} , 4\sqrt{3} , 3\sqrt{\dfrac{6}{5}} \\[1.5em] \text{(ii)} & \dfrac{5}{\sqrt{3}} , {\dfrac{7}{3}}{\sqrt{2}} , -\sqrt{3} , 3\sqrt{5} , 2\sqrt{7}. \\[1.5em] \end{matrix}$

Rationalise the denominator of the following :

$\begin{matrix} \text{(i)} & \dfrac{3}{4\sqrt{5}} \\[1.5em] \text{(ii)} & \dfrac{5\sqrt{7}}{\sqrt{3}} \\[1.5em] \text{(iii)} & \dfrac{3}{4 - \sqrt{7}} \\[1.5em] \text{(iv)} & \dfrac{17}{3\sqrt{2} + 1} \\[1.5em] \text{(v)} & \dfrac{16}{\sqrt{41}-5} \\[1.5em] \text{(vi)} & \dfrac{1}{\sqrt{7} - \sqrt{6}} \\[1.5em] \text{(vii)} & \dfrac{1}{\sqrt{5} + \sqrt{2}} \\[1.5em] \text{(viii)} & \dfrac{\sqrt{2} + \sqrt{3}}{\sqrt{2} - \sqrt{3}} \\[1.5em] \end{matrix}$

Simplify each of the following by rationalising the denominator:

$\begin{matrix} \text{(i)} & \dfrac{7 + 3\sqrt{5}}{7 - 3\sqrt{5}} \\[1.5em] \text{(ii)} & \dfrac{3 - 2\sqrt{2}}{3 + 2\sqrt{2}} \\[1.5em] \text{(iii)} & \dfrac{5 - 3\sqrt{14}}{7 + 2\sqrt{14}} \\[1.5em] \end{matrix}$