Simplify each of the following by rationalising the denominator:

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

$\text(i)$

$\dfrac{7 + 3\sqrt{5}}{7 - 3\sqrt{5}}$

Let us rationalise the denominator,

Then,

$\dfrac{7 + 3\sqrt{5}}{7 - 3\sqrt{5}} = \dfrac{7 + 3\sqrt{5}}{7 - 3\sqrt{5}} × \dfrac{7 + 3\sqrt{5}}{7 + 3\sqrt{5}} \\[1.5em] \Rightarrow\dfrac{(7 + 3\sqrt{5})^2}{7^2 - (3\sqrt{5})^2} \\[1.5em] \Rightarrow\dfrac{7^2 + (3\sqrt{5})^2 + 2 × 7 × 3\sqrt{5}}{49 - 45} \\[1.5em] \Rightarrow\dfrac{49 + 45 + 42\sqrt{5}}{49 - 45} \\[1.5em] \Rightarrow\dfrac{94 + 42\sqrt{5}}{49 - 45} \\[1.5em] \Rightarrow\dfrac{2 × (47 + 21\sqrt{5})}{4} \\[1.5em] \bold{\Rightarrow\dfrac{(47 + 21\sqrt{5})}{2}} \\[1.5em]$

$\text(ii)$

$\dfrac{3 - 2\sqrt{2}}{3 + 2\sqrt{2}}$

$\dfrac{3 - 2\sqrt{2}}{3 + 2\sqrt{2}} = \dfrac{3 - 2\sqrt{2}}{3 + 2\sqrt{2}} × \dfrac{3 - 2\sqrt{2}}{3 - 2\sqrt{2}} \\[1.5em] \Rightarrow\dfrac{(3 - 2\sqrt{2})^2}{3^2 - (2\sqrt{2})^2} \\[1.5em] \Rightarrow\dfrac{3^2 + (2\sqrt{2})^2 - 2 × 3 × 2\sqrt{2}}{9-8} \\[1.5em] \Rightarrow\dfrac{9 + 8 - 12\sqrt{2}}{1} \\[1.5em] \Rightarrow\dfrac{(17 - 12\sqrt{2})}{1} \\[1.5em] \bold{\Rightarrow{17 - 12\sqrt{2}}} \\[1.5em]$

$\text(iii)$

$\dfrac{5 - 3\sqrt{14}}{7 + 2\sqrt{14}}$

$\dfrac{5 - 3\sqrt{14}}{7 + 2\sqrt{14}} = \dfrac{5 - 3\sqrt{14}}{7 + 2\sqrt{14}} × \dfrac{7 - 2\sqrt{14}}{7 - 2\sqrt{14}} \\[1.5em] \Rightarrow\dfrac{(5 - 3\sqrt{14})(7 - 2\sqrt{14})}{7^2 - (2\sqrt{14})^2} \\[1.5em] \Rightarrow\dfrac{35 - 10\sqrt{14} - 21\sqrt{14} + (6 × 14)}{49 - 56} \\[1.5em] \Rightarrow\dfrac{119 - 31\sqrt{14}}{-7} \\[1.5em] \bold{\Rightarrow\dfrac{-119+31\sqrt{14}}{7}} \\[1.5em]$