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X=2+^3 find value of ^x+1/^x

Answer»
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} \\ = \frac{2 - \sqrt{3} }{1} = 2 - \sqrt{3} \\ x + \frac{1}{x} = 2 + \sqrt{3} + 2 - \sqrt{3} = 4 \\ {( \sqrt{x} + \frac{1}{ \sqrt{x} } )}^{2} = x + \frac{1}{x} + 2 \times \sqrt{x } \times \frac{1}{ \sqrt{x} } \\ = > {( \sqrt{x} + \frac{1}{ \sqrt{x} }) }^{2} = 4 + 2 \\ = > \sqrt{x} + \frac{1}{ \sqrt{x} } = \sqrt{6}


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