Find the value
\[ x = \frac{2}{\sqrt{10} – \sqrt{8}}, \quad y = \frac{2}{\sqrt{10} + 2\sqrt{2}} \]
Solution:
\[ x = \frac{2}{\sqrt{10} – 2\sqrt{2}} \times \frac{\sqrt{10} + 2\sqrt{2}}{\sqrt{10} + 2\sqrt{2}} = \frac{2(\sqrt{10} + 2\sqrt{2})}{10 – 8} \]
\[ = \sqrt{10} + 2\sqrt{2} \]
\[ y = \frac{2}{\sqrt{10} + 2\sqrt{2}} \times \frac{\sqrt{10} – 2\sqrt{2}}{\sqrt{10} – 2\sqrt{2}} = \frac{2(\sqrt{10} – 2\sqrt{2})}{10 – 8} \]
\[ = \sqrt{10} – 2\sqrt{2} \]
\[ x – y = (\sqrt{10} + 2\sqrt{2}) – (\sqrt{10} – 2\sqrt{2}) = 4\sqrt{2} \]
\[ (x – y)^2 = (4\sqrt{2})^2 = 32 \]