Упрощение алгебраических выражений. Задача 31

ОТВЕТ: 6.

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

\( = \dfrac{{2b}}{{{b^2}-a}} \cdot {\left( {\dfrac{1}{{3{a^2}b}}:\dfrac{{{b^2}-a}}{{{a^2}{b^2}}}} \right)^{-1}} = \dfrac{{2b}}{{{b^2}-a}} \cdot {\left( {\dfrac{1}{{3{a^2}b}} \cdot \dfrac{{{a^2}{b^2}}}{{{b^2}-a}}} \right)^{-1}} = \)

\( = \dfrac{{2b}}{{{b^2}-a}} \cdot {\left( {\dfrac{b}{{3\left( {{b^2}-a} \right)}}} \right)^{-1}} = \dfrac{{2b}}{{{b^2}-a}} \cdot \dfrac{{3\left( {{b^2}-a} \right)}}{b} = 6.\)

Ответ:  6.