Question

Find a and b if 7+ root 5 / 3- root 5 = a+b root 5

Answer 1

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$\frac{7 + \sqrt{5} }{3 - \sqrt{5} } = a + b \sqrt{5} \\ \to \frac{7 + \sqrt{5} }{3 - \sqrt{5} } \times \frac{3 + \sqrt{5} }{3 + \sqrt{5} } = a + b \sqrt{5} \\ \to \frac{21 + 7 \sqrt{5} + 3 \sqrt{5} + 5}{9 - 5} = a + b \sqrt{5} \\ \to \: \frac{26 + 10 \sqrt{5} }{4} = a + b \sqrt{5} \\ \to \: \frac{13}{2} + \frac{5 \sqrt{5} }{2} = a + b \sqrt{5} \\ comparing \: \: both \: \: sides \: we \: get \\ a = \frac{13}{2} \: \: \: b = \frac{5}{2}$