Write the following numbers in ascending order:

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

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

$3\sqrt{2} = \sqrt{9} × \sqrt{2} = \sqrt{9 × 2} = \sqrt{18} \\[1.5em] 2\sqrt{3} = \sqrt{4} × \sqrt{3} = \sqrt{4 × 3} = \sqrt{12} \\[1.5em] \sqrt{15} = \sqrt{15} \\[1.5em] 4 = \sqrt{16} \\[1.5em] \text{Since} , 12 \lt 15 \lt 16 \lt 18 \\[1.5em] \Rightarrow \sqrt{12} \lt \sqrt{15} \lt \sqrt{16} \lt \sqrt{18} \\[1.5em] \Rightarrow 2\sqrt3 \lt \sqrt{15} \lt 4 \lt 3\sqrt{2}$

Hence, the given numbers in ascending order are, $2\bold{\sqrt{3}} , \bold{\sqrt{15}} , \bold{4} , \bold{3\sqrt{2}}$.

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

$3\sqrt{2} = \sqrt{9} × \sqrt{2} = \sqrt{9 × 2} = \sqrt{18} \\[1.5em] 2\sqrt{8} = \sqrt{4} × \sqrt{8} = \sqrt{4 × 8} = \sqrt{32} \\[1.5em] 4 = \sqrt{16} \\[1.5em] \sqrt{50} = \sqrt{50} \\[1.5em] 4\sqrt{3} = \sqrt{16} × \sqrt{3} = \sqrt{16 × 3} = \sqrt{48} \\[1.5em] \text{Since} , 16 \lt 18 \lt 32 \lt 48 \lt 50 \\[1.5em] \sqrt{16} \lt \sqrt{18} \lt \sqrt{32} \lt \sqrt{48} \lt \sqrt{50} \\[1.5em] \Rightarrow 4 \lt 3\sqrt2 \lt 2\sqrt{8} \lt 4\sqrt{3} \lt \sqrt{50}$

Hence , $3\bold{\sqrt{2}} , 2\bold{\sqrt{8}} , 4\bold{\sqrt{3}} , 4\bold{\sqrt{50}}$ are in ascending order .