If $$x = 3 + 2\sqrt 2 ,$$    then the value of $$\left( {\sqrt x - \frac{1}{{\sqrt x }}} \right)$$   is:

$$\eqalign{ & {\left( {\sqrt x - \frac{1}{{\sqrt x }}} \right)^2} \cr & = x + \frac{1}{x} - 2 \cr & = \left( {3 + 2\sqrt 2 } \right) + \frac{1}{{\left( {3 + 2\sqrt 2 } \right)}} - 2 \cr & = \left( {3 + 2\sqrt 2 } \right) + \frac{1}{{\left( {3 + 2\sqrt 2 } \right)}} \times \frac{{\left( {3 - 2\sqrt 2 } \right)}}{{\left( {3 - 2\sqrt 2 } \right)}} - 2 \cr & = \left( {3 + 2\sqrt 2 } \right) + \left( {3 - 2\sqrt 2 } \right) - 2 \cr & = 4 \cr & \therefore \left( {\sqrt x - \frac{1}{{\sqrt x }}} \right) = 2 \cr} $$