Determine rational numbers \(a\) and \(b\)
\[ \frac{4 + \sqrt{2}}{2 + \sqrt{2}} = a – \sqrt{b} \]
Solution:
\[ \frac{4 + \sqrt{2}}{2 + \sqrt{2}} \times \frac{2 – \sqrt{2}}{2 – \sqrt{2}} \]
\[ = \frac{(4 + \sqrt{2})(2 – \sqrt{2})}{(2)^2 – (\sqrt{2})^2} \]
\[ = \frac{8 – 4\sqrt{2} + 2\sqrt{2} – 2}{4 – 2} \]
\[ = \frac{6 – 2\sqrt{2}}{2} \]
\[ = 3 – \sqrt{2} \]
Comparing with \(a – \sqrt{b}\)
\[ a = 3, \quad b = 2 \]