Write the following real numbers in descending order:

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

(i) Write all the numbers as square root under one radical :

$\dfrac{9}{\sqrt{2}} = \dfrac{\sqrt{81}}{\sqrt{2}} = \sqrt{\dfrac{81}{2}} = \sqrt{40.5} \\[1.5em] {\dfrac{3}{2}}{\sqrt{5}} = \sqrt{\dfrac{9}{4}}×{\sqrt{5}} = \sqrt{\dfrac{9 × 5}{4}} = \sqrt{\dfrac{45}{4}} = \sqrt{11.25} \\[1.5em] 4\sqrt{3} = \sqrt{16} × \sqrt{3} = \sqrt{16 × 3} = \sqrt{48} \\[1.5em] 3\sqrt{\dfrac{6}{5}} = \sqrt{9} × \sqrt{\dfrac{6}{5}} = \sqrt{\dfrac{9 × 6}{5}} = \sqrt{\dfrac{54}{5}} = \sqrt{10.8} \\[1.5em] \text{Since} , 48 \gt 40.5 \gt 11.25 \gt 10.8 \\[1.5em] \Rightarrow \sqrt{48} \gt \sqrt{40.5} \gt \sqrt{11.25} \gt \sqrt{10.8} \\[1.5em] \Rightarrow 4\sqrt3 \gt \dfrac{9}{\sqrt{2}} \gt {\dfrac{3}{2}}{\sqrt{5}} \gt 3\sqrt{\dfrac{6}{5}}$

Hence, the given numbers in descending order are $\bold{4\sqrt{3}} , \bold{\dfrac{9}{\sqrt{2}}} , \bold{{\dfrac{3}{2}}{\sqrt{5}}} , \bold{3\sqrt{\dfrac{6}{5}}}$.

(ii) Write all the numbers as square root under one radical :

$\dfrac{5}{\sqrt{3}} = \dfrac{\sqrt{25}}{\sqrt{3}} = \sqrt{\dfrac{25}{3}} = \sqrt{8.33} \\[1.5em] {\dfrac{7}{3}}{\sqrt{2}} = \sqrt{\dfrac{49}{9}} ×{\sqrt{2}} = \sqrt{\dfrac{49 × 2}{9}} = \sqrt{\dfrac{98}{9}} = \sqrt{10.88} \\[1.5em] -\sqrt{3} = -\sqrt{3} \\[1.5em] 3\sqrt{5} = \sqrt{9 × 5} = \sqrt{45} \\[1.5em] 2\sqrt{7} = \sqrt{4 × 7} = \sqrt{28} \\[1.5em] \text{Since} , 45 \gt 28 \gt 10.88 \gt 8.33 \gt -3 \\[1.5em] \sqrt{45} \gt \sqrt{28} \gt \sqrt{10.88} \gt \sqrt{8.33} \gt -\sqrt{3} \\[1.5em] \Rightarrow 3\sqrt{5} \gt 2\sqrt{7} \gt {\dfrac{7}{3}}{\sqrt{2}} \gt \dfrac{5}{\sqrt{3}} \gt -\sqrt{3}$

Hence, $\bold{3\sqrt{5}}, \bold{2\sqrt{7}}, \bold{{{\dfrac{7}{3}}{\sqrt{2}}}}, \bold{\dfrac{5}{\sqrt{3}}}, -\bold{\sqrt{3}}$ are in descending order.