Question

IF
$x = 3 + 2 \sqrt{2}$
Then find the value of
$\sqrt{x} - \frac{1}{ \sqrt{x} }$

Answer 1

$x = 3 + 2 \sqrt{2 } \\ \sqrt{x} - \frac{1}{ \sqrt{x} } \\ = \frac{x - 1}{ \sqrt{x} } \\ = \frac{3 + 2 \sqrt{2 } - 1 }{ \sqrt{3 + 2 \sqrt{2} } } \\ = \frac{2 + 2 \sqrt{2} }{ \sqrt{3 + 2 \sqrt{2} } } \times \frac{ \sqrt{3 - 2 \sqrt{2} } }{ \sqrt{3 - 2 \sqrt{2} } } \\ = \frac{2(1 + \sqrt{2} )( \sqrt{3 - 2 \sqrt{2} }) }{ \sqrt{9 - 8} } \\ = 2(1 + \sqrt{2} )( \sqrt{3 + 2 \sqrt{2} } )$

Answer 2

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