Question

questions are given above in the picture

Answer 1

$x = 2 + \sqrt{3} \\ \frac{1}{x} = \frac{1}{2 + \sqrt{3} } = \frac{2 - \sqrt{3} }{(2 + \sqrt{3} )(2 - \sqrt{3} )} \\ = \frac{2 - \sqrt{3} }{ {2}^{2} - {( \sqrt{3}) }^{2} } = \frac{2 - \sqrt{3} }{4 - 3} = 2 - \sqrt{3} \\ \\ now... \\ (x + \frac{1}{x} )^{3} = (2 + \sqrt{3} + 2 - \sqrt{3} ) ^{3} \\ = (4) ^{3} = 64$
$\\ x = 5 - 2 \sqrt{6} \\ \frac{1}{x} = \frac{1}{5 - 2 \sqrt{6} } = \frac{5 + 2 \sqrt{6} }{(5 - 2 \sqrt{6})(5 + 2 \sqrt{6}) } \\ = \frac{5 + 2 \sqrt{6} }{ {5}^{2} - {(2 \sqrt{6}) }^{2} } = \frac{5 + 2 \sqrt{6} }{25 - 24} \\ = 5 + 2 \sqrt{6 } \\ \\ now... \\ ( {x}^{2} + \frac{ 1 }{ {x}^{2} } ) = (x + \frac{1}{x} )^{2} - 2x. \frac{1}{x} \\ = (5 - 2 \sqrt{6} + 5 + 2 \sqrt{6} )^{2} - 2 \\ = {(10)}^{2} - 2 \\ = 100 - 2 \\ = 98$
Hope this is ur required answer
Proud to help you