Rút gọn biểu thức \[P = \left( {\sqrt {ab} - \frac{{ab}}{{a + \sqrt {ab} }}} \right):\frac{{\sqrt[4]{{ab}} - \sqrt b }}{{a - b}}\left( {a > 0,b > 0,a \ne b} \right)\] ta được kết quả là:

\[P = \left( {\sqrt {ab} - \frac{{ab}}{{a + \sqrt {ab} }}} \right):\frac{{\sqrt[4]{{ab}} - \sqrt b }}{{a - b}}\]

\[ = \left( {\frac{{\sqrt {ab} \left( {a + \sqrt {ab} } \right) - ab}}{{a + \sqrt {ab} }}} \right).\frac{{a - b}}{{\sqrt[4]{{ab}} - {{\left( {\sqrt[4]{b}} \right)}^2}}}\]

\[ = \frac{{a.\sqrt {ab} + ab - ab}}{{{{\left( {\sqrt a } \right)}^2} + \sqrt a .\sqrt b }}.\frac{{\left( {\sqrt a - \sqrt b } \right)\left( {\sqrt a + \sqrt b } \right)}}{{\sqrt[4]{b}\left( {\sqrt[4]{a} - \sqrt[4]{b}} \right)}}\]

\[ = \frac{{a\sqrt {ab} }}{{\sqrt a \left( {\sqrt a + \sqrt b } \right)}}.\frac{{\left( {\sqrt a - \sqrt b } \right)\left( {\sqrt a + \sqrt b } \right)}}{{\sqrt[4]{b}\left( {\sqrt[4]{a} - \sqrt[4]{b}} \right)}} = \frac{{a\sqrt a .\sqrt b }}{{\sqrt a }}.\frac{{{{\left( {\sqrt[4]{a}} \right)}^2} - {{\left( {\sqrt[4]{b}} \right)}^2}}}{{\sqrt[4]{b}\left( {\sqrt[4]{a} - \sqrt[4]{b}} \right)}}\]

\[ = \frac{{a\sqrt b .\left( {\sqrt[4]{a} - \sqrt[4]{b}} \right)\left( {\sqrt[4]{a} + \sqrt[4]{b}} \right)}}{{\sqrt[4]{b}\left( {\sqrt[4]{a} - \sqrt[4]{b}} \right)}} = \frac{{a{{\left( {\sqrt[4]{b}} \right)}^2}.\left( {\sqrt[4]{a} + \sqrt[4]{b}} \right)}}{{\sqrt[4]{b}}} = a\sqrt[4]{b}\left( {\sqrt[4]{a} + \sqrt[4]{b}} \right)\]

Vậy \[P = a\sqrt[4]{b}(\sqrt[4]{a} + \sqrt[4]{b}).\]