Question

Find a and b when 7 + √5/7-√5= a + b √5

Answer 1

$\large \green{ \underline{ \underline{explanation}}} \\ \\ \large \implies{ \frac{7 + \sqrt{5} }{7 - \sqrt{5} } } = a + b \sqrt{5} \\ \\ \large \implies{rationalise \: the \: \: equation} \\ \\ \large \implies{ \frac{7 + \sqrt{5} }{7 - \sqrt{5} } } \times \frac{7 + \sqrt{5} }{7 + \sqrt{5} } \\ \\ \large \implies{ \frac{(7 + \sqrt{5}) {}^{2} }{49 - 5} } \\ \\ \large \implies{ \frac{49 + 5 + 14 \sqrt{5} }{44} } \\ \\ \large \implies{ \frac{54 + 14 \sqrt{5} }{44} } = a + b \sqrt{5} \\ \\ \large \implies{a \: = \frac{54}{44} } \: \: and \: \: b \: = \frac{14}{44} \\ \\ \large \implies{ \boxed{ a \: = \frac{27}{22} }} \\ \\ \large \implies{ \boxed{b \: = \frac{7}{22} }}$