[tex] \sqrt{243 \sqrt{81 \sqrt[3]{3} } }= 9\sqrt{3}( \sqrt{9 \sqrt[3]{x} } )=9\sqrt{3}\sqrt{9}( 3\sqrt[6]{3})=27\sqrt{3}\sqrt[6]{3}=(3^{ \frac{1}{6}+3+\frac{1}{2}}) [/tex]
[tex]log_{3} (3^{\frac{1}{6]+3+\frac{1}{2}} )=log_{3} (3^ {\frac{1}{6}+3+\frac{1}{2}} )= \frac{22}{6}....(1)[/tex]
[tex]log_{2} \sqrt[4]{64}=log_{2} \sqrt[4]{ 2^6}= log_{2} 2^{ \frac{3}{2} }= \frac{3}{2}....(2)[/tex]
[tex]log_{e} e^{-10}=-10[/tex]
[tex]log_{2} ( \sqrt[4]{64} )+log_{e}(e^{-10})= \frac{3}{2}-10 [/tex]
[tex] \frac{log_{3}(\sqrt{243 \sqrt{81 \sqrt[3]{3} } }) }{log_{2} ( \sqrt[4]{64} )+log_{e}(e^{-10})} [/tex]
[tex]= \frac{\frac{22}{6}}{\frac{3}{2}-10} [/tex]
[tex]= \frac{ \frac{22}{6}} { \frac{-17}{2} }= \frac{22}{(-17)(3)}= \frac{-22}{51} [/tex]
