Evaluate, correct to one place of decimal, the expression $\dfrac{5}{\sqrt{20} - \sqrt{10}}, \text{ if } \sqrt{5} = 2.2 \text{ and } \sqrt{10} = 3.2$

Solving,

$\Rightarrow \dfrac{5}{\sqrt{20} - \sqrt{10}} \\[1em] \Rightarrow \dfrac{5}{2\sqrt{5} - \sqrt{10}} \\[1em] \Rightarrow \dfrac{5}{2 \times 2.2 - 3.2} \\[1em] \Rightarrow \dfrac{5}{4.4 - 3.2} \\[1em] \Rightarrow \dfrac{5}{1.2} \\[1em] \Rightarrow \dfrac{50}{12} \\[1em] \Rightarrow \dfrac{25}{6} = 4.2$

Another method of solving is by rationalizing,

$\Rightarrow \dfrac{5}{\sqrt{20} - \sqrt{10}} \times \dfrac{\sqrt{20} + \sqrt{10}}{\sqrt{20} + \sqrt{10}} \\[1em] \Rightarrow \dfrac{5(\sqrt{20} + \sqrt{10})}{(\sqrt{20})^2 - (\sqrt{10})^2} \\[1em] \Rightarrow \dfrac{5(\sqrt{20} + \sqrt{10})}{20 - 10} \\[1em] \Rightarrow \dfrac{5(\sqrt{20} + \sqrt{10})}{10} \\[1em] \Rightarrow \dfrac{(2\sqrt{5} + \sqrt{10})}{2} \\[1em] \Rightarrow \dfrac{2 \times 2.2 + 3.2}{2} \\[1em] \Rightarrow \dfrac{4.4 + 3.2}{2} \\[1em] \Rightarrow \dfrac{7.6}{2} \\[1em] \Rightarrow 3.8$

Hence, $\dfrac{\sqrt{x^2 + y^2} - y}{x - \sqrt{x^2 - y^2}} ÷ \dfrac{\sqrt{x^2 - y^2} + x}{\sqrt{x^2 + y^2} + y}$ = 3.8 or 4.2