Упрощение алгебраических выражений. Задача 39

ОТВЕТ: 5.

\(\left( {\dfrac{{a-4b}}{{a + \sqrt {ab} -6b}}-\dfrac{{a-9b}}{{a + 6\sqrt {ab}  + 9b}}} \right)\,\dfrac{{{b^{-\frac{1}{2}}}\,\left( {a-9b} \right)}}{{\sqrt a -3{b^{\frac{1}{2}}}}} = \)

\( = \left( {\dfrac{{{{\left( {\sqrt a } \right)}^2}-{{\left( {2\sqrt b } \right)}^2}}}{{{{\left( {\sqrt a } \right)}^2} + 3\sqrt {ab} -2\sqrt {ab} -6{{\left( {\sqrt b } \right)}^2}}}-\dfrac{{{{\left( {\sqrt a } \right)}^2}-{{\left( {3\sqrt b } \right)}^2}}}{{{{\left( {\sqrt a } \right)}^2} + 2 \cdot \sqrt a  \cdot 3\sqrt b  + {{\left( {3\sqrt b } \right)}^2}}}} \right) \cdot \frac{{{{\left( {\sqrt a } \right)}^2}-{{\left( {3\sqrt b } \right)}^2}}}{{\sqrt b \left( {\sqrt a -3\sqrt b } \right)}} = \)

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

\( = \left( {\dfrac{{\left( {\sqrt a -2\sqrt b } \right)\left( {\sqrt a  + 2\sqrt b } \right)}}{{\left( {\sqrt a  + 3\sqrt b } \right)\left( {\sqrt a -2\sqrt b } \right)}}-\dfrac{{\sqrt a -3\sqrt b }}{{\sqrt a  + 3\sqrt b }}} \right) \cdot \dfrac{{\sqrt a  + 3\sqrt b }}{{\sqrt b }} = \)

\( = \left( {\dfrac{{\sqrt a  + 2\sqrt b }}{{\sqrt a  + 3\sqrt b }}-\dfrac{{\sqrt a -3\sqrt b }}{{\sqrt a  + 3\sqrt b }}} \right) \cdot \dfrac{{\sqrt a  + 3\sqrt b }}{{\sqrt b }} = \)

\( = \dfrac{{\sqrt a  + 2\sqrt b -\sqrt a  + 3\sqrt b }}{{\sqrt a  + 3\sqrt b }} \cdot \dfrac{{\sqrt a  + 3\sqrt b }}{{\sqrt b }} = \dfrac{{5\sqrt b }}{{\sqrt b }} = 5.\)

Ответ:  5.