Determine rational numbers \(a\) and \(b\)
\[ \frac{3 + \sqrt{2}}{3 – \sqrt{2}} = a + b\sqrt{2} \]
Solution:
\[ \frac{3 + \sqrt{2}}{3 – \sqrt{2}} \times \frac{3 + \sqrt{2}}{3 + \sqrt{2}} \]
\[ = \frac{(3 + \sqrt{2})^2}{3^2 – (\sqrt{2})^2} \]
\[ = \frac{9 + 6\sqrt{2} + 2}{9 – 2} \]
\[ = \frac{11 + 6\sqrt{2}}{7} \]
\[ = \frac{11}{7} + \frac{6}{7}\sqrt{2} \]
Comparing with \(a + b\sqrt{2}\)
\[ a = \frac{11}{7}, \quad b = \frac{6}{7} \]