Rút gọn biểu thức: \(P = \left( {\frac{{x\sqrt x - 1}}{{x - \sqrt x }} - \frac{{x\sqrt x + 1}}{{x + \sqrt x }}} \right):\left[ {\frac{{2\left( {x - 2\sqrt x + 1} \right)}}{{x - 1}}} \right]\) (với \(x > 0\) và \(x \ne 1\) )

Điều kiện: \(x > 0,\;\;x \ne 1.\)

\(\begin{array}{l}P = \left( {\frac{{x\sqrt x  - 1}}{{x - \sqrt x }} - \frac{{x\sqrt x  + 1}}{{x + \sqrt x }}} \right):\left[ {\frac{{2\left( {x - 2\sqrt x  + 1} \right)}}{{x - 1}}} \right]\\ = \left[ {\frac{{{{\left( {\sqrt x } \right)}^3} - {1^3}}}{{\sqrt x \left( {\sqrt x  - 1} \right)}} - \frac{{{{\left( {\sqrt x } \right)}^3} + {1^3}}}{{\sqrt x \left( {\sqrt x  + 1} \right)}}} \right]:\left[ {\frac{{2{{\left( {\sqrt x  - 1} \right)}^2}}}{{{{\left( {\sqrt x } \right)}^2} - 1}}} \right]\\ = \left[ {\frac{{\left( {\sqrt x  - 1} \right)\left( {x + \sqrt x  + 1} \right)}}{{\sqrt x \left( {\sqrt x  - 1} \right)}} - \frac{{\left( {\sqrt x  + 1} \right)\left( {x - \sqrt x  + 1} \right)}}{{\sqrt x \left( {\sqrt x  + 1} \right)}}} \right]:\left[ {\frac{{2{{\left( {\sqrt x  - 1} \right)}^2}}}{{\left( {\sqrt x  - 1} \right)\left( {\sqrt x  + 1} \right)}}} \right]\\ = \left( {\frac{{\left( {x + \sqrt x  + 1} \right)}}{{\sqrt x }} - \frac{{\left( {x - \sqrt x  + 1} \right)}}{{\sqrt x }}} \right):\frac{{2\left( {\sqrt x  - 1} \right)}}{{\sqrt x  + 1}}\\ = \frac{{2\sqrt x }}{{\sqrt x }}.\frac{{\sqrt x  + 1}}{{2\left( {\sqrt x  - 1} \right)}} = \frac{{\sqrt x  + 1}}{{\sqrt x  - 1}}\end{array}\)

Chọn D.