Question

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Answer 1

$given \\ \frac{ \sqrt{3} - 1 }{ \sqrt{3} + 1 } = a - b \sqrt{3} \\ \frac{ \sqrt{3} - 1 \times \sqrt{3} - 1 }{ \sqrt{3} + 1 \times \sqrt{3} - 1 } = a - b \sqrt{3} \\ by \: formula \\ \:( a + b) \: (a - b) = a {}^{2} - b {}^{2} \\ and \:( a - b) \: (a - b) = (a {}^{2} + b {}^{2} - 2ab ) \\ \frac{3 + 1 - 2 \sqrt{3} }{3 - 1} \\ = \frac{4 - 2 \sqrt{3} }{2} \\ = 2 - \sqrt{3} = a - b \sqrt{3} \\ by \: comparing \: both \: sides \: a = 2 \: and \: b = 1$