Determine rational numbers \(a\) and \(b\)
\[ \frac{4 + 3\sqrt{5}}{4 – 3\sqrt{5}} = a + b\sqrt{5} \]
Solution:
\[ \frac{4 + 3\sqrt{5}}{4 – 3\sqrt{5}} \times \frac{4 + 3\sqrt{5}}{4 + 3\sqrt{5}} \]
\[ = \frac{(4 + 3\sqrt{5})^2}{4^2 – (3\sqrt{5})^2} \]
\[ = \frac{16 + 24\sqrt{5} + 45}{16 – 45} \]
\[ = \frac{61 + 24\sqrt{5}}{-29} \]
\[ = -\frac{61}{29} – \frac{24}{29}\sqrt{5} \]
Comparing with \(a + b\sqrt{5}\)
\[ a = -\frac{61}{29}, \quad b = -\frac{24}{29} \]