Question

Rationalise the denominator of 4 upon 2+√3+√7

Answer 1

$\textbf{ANSWER :}$

$Now, \: \: \frac{4}{2 + \sqrt{3} + \sqrt{7} } \\ \\ = \frac{4}{(2 + \sqrt{3} ) + \sqrt{7} } \\ \\ = \frac{4((2 + \sqrt{3} ) - \sqrt{7} )}{((2 + \sqrt{3} ) + \sqrt{7} )((2 + \sqrt{3}) - \sqrt{7} )} \\ \\ = \frac{4((2 + \sqrt{3}) - \sqrt{7} ) }{ {(2 + \sqrt{3} )}^{2} - {( \sqrt{7} )}^{2} } \\ \\ = \frac{4((2 + \sqrt{3} ) - \sqrt{7}) }{4 + 4 \sqrt{3} + 3 - 7} \\ \\ = \frac{4((2 + \sqrt{3} ) - \sqrt{7} )}{4 \sqrt{3} } \\ \\ = \frac{2 + \sqrt{3} - \sqrt{7} }{ \sqrt{3} } \\ \\ = \frac{ \sqrt{3}(2 + \sqrt{3} - \sqrt{7} ) }{ \sqrt{3} \times \sqrt{3} } \\ \\ = \frac{ \sqrt{3} (2 + \sqrt{3} - \sqrt{7} )}{3} \\ \\=\frac{2\sqrt3 + 3 - \sqrt21}{3}, \\ \\ which \: \: is \: \: the \: \: required \: \: answer.$

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