Question

find the value of a and b is 3 + root 7 upon 3 minus 4 root 7 is equal to a + b root 7

Answer 1

Identity used :

$(a + b)(a - b) = {a}^{2} - {b}^{2}$

$\frac{3 + \sqrt{7} }{3 - 4 \sqrt{7} } = a + b \sqrt{7} \\ \\ \bf \: L.H.S \\ \\ \bf \: On \: rationalizing \: the \: denominator \\ \bf \: we \: get \\ \\ = \frac{3 + \sqrt{7} }{3 - 4 \sqrt{7} } \times \frac{3 + 4 \sqrt{7} }{3 + 4 \sqrt{7} } \\ \\ = \frac{3(3 + 4 \sqrt{7}) + \sqrt{7} (3 + 4 \sqrt{7} ) }{ {( {3)}^{2} } - {(4 \sqrt{7} )}^{2} } \\ \\ = \frac{9 + 12 \sqrt{7} + 3 \sqrt{7} + 28 }{9 - 112} \\ \\ = \frac{37 + 15 \sqrt{7} }{ - 103} \\ \\ = \frac{ - 37 - 15 \sqrt{7} }{103} \\ \\ \bf \: On \: comparing \: L.H.S \: and \: R.H.S \\ \bf \: we \: get \\ \\ \frac{ - 37}{103} + \frac{ - 15 \sqrt{7} }{103} = a + b \sqrt{7} \\ \\ \bf \: a \: = \frac{ - 37}{103} \: \: : b = \frac{ - 15}{103}$