\(\dfrac { \left(^3 \log 36 \right)^2 - \left(^3 \log 4 \right)^2 }{ ^3 \log \sqrt{12}} = \dotso\)
\begin{equation*}
\begin{split}
& \frac { \left(^3 \log 36 \right)^2 - \left(^3 \log 4 \right)^2 }{ ^3 \log \sqrt{12}} \\\\
& \frac { \left(^3 \log 36 + \: ^3 \log 4 \right) \left(^3 \log 36 - \: ^3 \log 4 \right)}{ ^3 \log (12)^{\frac 12}} \\\\
& \frac {^3 \log (36 \:.\: 4) \:.\: ^3 \log \frac {36}{4}}{\frac 12 \:.\: ^3 \log 12} \\\\
& \frac {^3 \log 144 \:.\: ^3 \log 9}{\frac 12 \:.\: ^3 \log 12} \\\\
& \frac {^3 \log 12^2 \:.\: ^3 \log 3^2}{\frac 12 \:.\: ^3 \log 12} \\\\
& \frac {2 \:.\: ^3 \log 12 \:.\: 2 \:.\: ^3 \log 3}{\frac 12 \:.\: ^3 \log 12} \\\\
& \frac {4 \:.\: \cancel {^3 \log 12}}{\frac 12 \:.\: \cancel {^3 \log 12}} \\\\
& 8
\end{split}
\end{equation*}