Запишем область допустимых значений на переменную a: \(\left\{ {\begin{array}{*{20}{c}}{a > 0,\,\,\,\,\,\,\,\,\,\,}\\{a \ne 1,\,\,\,\,\,\,\,\,\,\,}\\{{a^2}-1 > 0}\end{array}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,a > 1.} \right.\)
\(\dfrac{{{{\log }_a}\sqrt {{a^2}-1} \cdot \log _{\frac{1}{a}}^2\sqrt {{a^2}-1} }}{{{{\log }_{{a^2}}}\left( {{a^2}-1} \right) \cdot {{\log }_{\sqrt[3]{a}}}\sqrt[6]{{{a^2}-1}}}} = \dfrac{{\dfrac{1}{2}{{\log }_a}\left( {{a^2}-1} \right) \cdot \log _{{a^{-1}}}^2{{\left( {{a^2}-1} \right)}^{\frac{1}{2}}}}}{{\dfrac{1}{2}{{\log }_a}\left( {{a^2}-1} \right) \cdot 3 \cdot \dfrac{1}{6}{{\log }_a}\left( {{a^2}-1} \right)}} = \)
\( = \dfrac{{2 \cdot \dfrac{1}{4}\log _a^2\left( {{a^2}-1} \right)}}{{{{\log }_a}\left( {{a^2}-1} \right)}} = \dfrac{1}{2}{\log _a}\left( {{a^2}-1} \right) = {\log _a}\sqrt {{a^2}-1} ,\,\,\,\,\,\,\,a > 1.\)
Ответ: \({\log _a}\sqrt {{a^2}-1} \), где \(a > 1.\)