Determine rational numbers \(a\) and \(b\)
\[ \frac{\sqrt{11} – \sqrt{7}}{\sqrt{11} + \sqrt{7}} = a – b\sqrt{77} \]
Solution:
\[ \frac{\sqrt{11} – \sqrt{7}}{\sqrt{11} + \sqrt{7}} \times \frac{\sqrt{11} – \sqrt{7}}{\sqrt{11} – \sqrt{7}} \]
\[ = \frac{(\sqrt{11} – \sqrt{7})^2}{11 – 7} \]
\[ = \frac{11 + 7 – 2\sqrt{77}}{4} \]
\[ = \frac{18 – 2\sqrt{77}}{4} \]
\[ = \frac{9}{2} – \frac{1}{2}\sqrt{77} \]
Comparing with \(a – b\sqrt{77}\)
\[ a = \frac{9}{2}, \quad b = \frac{1}{2} \]