Determine rational numbers \(a\) and \(b\)
\[ \frac{\sqrt{3} – 1}{\sqrt{3} + 1} = a – b\sqrt{3} \]
Solution:
\[ \frac{\sqrt{3} – 1}{\sqrt{3} + 1} \times \frac{\sqrt{3} – 1}{\sqrt{3} – 1} \]
\[ = \frac{(\sqrt{3} – 1)^2}{(\sqrt{3})^2 – (1)^2} \]
\[ = \frac{3 – 2\sqrt{3} + 1}{3 – 1} \]
\[ = \frac{4 – 2\sqrt{3}}{2} \]
\[ = 2 – \sqrt{3} \]
Comparing with \(a – b\sqrt{3}\)
\[ a = 2, \quad b = 1 \]