Упрощение алгебраических выражений. Задача 36

ОТВЕТ: \(ab\).

\(\left( {\sqrt {ab} -\dfrac{{ab}}{{a + \sqrt {ab} }}} \right)\,{\left( {\dfrac{{\sqrt[4]{{ab}}-\sqrt b }}{{a-b}}} \right)^{-1}}{\left( {\dfrac{{\sqrt[4]{a} + \sqrt[4]{b}}}{{\sqrt[4]{{{b^3}}}}}} \right)^{-1}} = \)

\( = \dfrac{{a\sqrt {ab}  + ab-ab}}{{{{\left( {\sqrt a } \right)}^2} + \sqrt a \sqrt b }} \cdot \dfrac{{{{\left( {\sqrt a } \right)}^2}-{{\left( {\sqrt b } \right)}^2}}}{{\sqrt[4]{a} \cdot \sqrt[4]{b}-{{\left( {\sqrt[4]{b}} \right)}^2}}} \cdot \dfrac{{\sqrt[4]{{{b^3}}}}}{{\sqrt[4]{a} + \sqrt[4]{b}}} = \)

\( = \dfrac{{a\sqrt a \sqrt b }}{{\sqrt a \left( {\sqrt a  + \sqrt b } \right)}} \cdot \dfrac{{\left( {\sqrt a -\sqrt b } \right)\left( {\sqrt a  + \sqrt b } \right)}}{{\sqrt[4]{b}\left( {\sqrt[4]{a}-\sqrt[4]{b}} \right)}} \cdot \dfrac{{{{\left( {\sqrt[4]{b}} \right)}^3}}}{{\sqrt[4]{a} + \sqrt[4]{b}}} = \)

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

Ответ:  \(ab.\)