Log trouble

$\begin{aligned} x & = \left(\frac 16\left(\frac {2\log_{10}1728}{1+\frac 12 \log_{10}0.36+\frac 13\log_{10}8}\right)^\frac 12 \right)^{-1} \\ & = \left(\frac 16\left(\frac {2\log_{10}1728}{\log_{10}10+ \log_{10}\sqrt{0.36}+\log_{10}\sqrt[3]8}\right)^\frac 12 \right)^{-1} \\ & = \left(\frac 16\left(\frac {2\log_{10}1728}{\log_{10}10+ \log_{10} 0.6 +\log_{10}2} \right)^\frac 12 \right)^{-1} \\ & = \left(\frac 16\left(\frac {2\log_{10}1728}{\log_{10}(10 \times 0.6 \times 2)} \right)^\frac 12 \right)^{-1} \\ & = \left(\frac 16\left(\frac {2\log_{10} 12^3}{\log_{10} 12} \right)^\frac 12 \right)^{-1} \\ & = \left(\frac 16\left(\frac {6 \cancel{\log_{10} 12}}{\cancel{\log_{10} 12}} \right)^\frac 12 \right)^{-1} \\ & = \left(\frac {\sqrt 6}6 \right)^{-1} = \left(\frac 1{\sqrt 6} \right)^{-1} = \sqrt 6 \end{aligned}$

Therefore, $n=\boxed{6}$ .