Question

If x=√3+√3/√3-√2, find x^2+1/x^2

Answer 1

Answer:

$x= \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } \\ \\ \frac{1}{x} = \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = {(x + \frac{1}{x}) }^{2} - 2 \\ \\ = {( \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } + \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } ) }^{2} - 2 \\ \\ = ( { {( \sqrt{3 } - \sqrt{2}) }^{2} + {( \sqrt{3} + \sqrt{2} )}^{2}) }^{2} - 2\\ \\ = {( 3+ 2 - 2 \sqrt{6} + 3 + 2 + 2 \sqrt{6} )}^{2} - 2 \\ \\ = 100 - 2 = 98$