Question

rationalize the denominator of 4 by 2 + root 3 + root 7
$4 \div 2 + \sqrt{3 + \sqrt{7} }$

Answer 1

$= \frac{4}{2 + \sqrt{3} + \sqrt{7} } \\ = \frac{4}{(2 + \sqrt{3}) + \sqrt{7} } \times \frac{(2 + \sqrt{3}) - \sqrt{7} }{(2 + \sqrt{3} ) - \sqrt{7} } \\ = \frac{8 + 4 \sqrt{3} - 4 \sqrt{7} }{4 + 3 + 4 \sqrt{3} - 7} \: \: \: \: \: \: by \: open \: identity \: in \: denominator \\ = \frac{8 + 4 \sqrt{3} - 4 \sqrt{7} }{4 \sqrt{3} } \\ = \frac{2 + \sqrt{3} - \sqrt{7} }{ \sqrt{3} } \times \frac{ \sqrt{3} }{ \sqrt{3} } \\ = \frac{2 \sqrt{3} + 3 - \sqrt{21} }{3} \\ = \frac{3 + 2 \sqrt{3} - \sqrt{21} }{3} \: \: \: \: \: ans$