Calculate $\;$ $\left[\left(128^{\frac{3}{7}} \times 27^{\frac{1}{3}} \times 10^{- \log 48}\right)^{\frac{-1}{2}} + \left(\cot \dfrac{2 \pi}{3}\right)^{-1}\right]^2 + 2 \times 6^{\frac{1}{2}}$
$\left[\left(128^{\frac{3}{7}} \times 27^{\frac{1}{3}} \times 10^{- \log 48}\right)^{\frac{-1}{2}} + \left(\cot \dfrac{2 \pi}{3}\right)^{-1}\right]^2 + 2 \times 6^{\frac{1}{2}}$
$= \left[\left(2^{7 \times \frac{3}{7}} \times 3^{3 \times \frac{1}{3}} \times 10^{\log \left(\frac{1}{48}\right)}\right)^{\frac{-1}{2}} + \dfrac{1}{\cot \dfrac{2 \pi}{3}}\right]^2 + 2 \sqrt{6}$
$\hspace{3cm}$ $\left[\text{Note: } \log n^m = m \log n\right]$
$= \left[\left(2^3 \times 3 \times \dfrac{1}{48}\right)^{\frac{-1}{2}} + \tan \dfrac{2 \pi}{3}\right]^2 + 2 \sqrt{6}$ $\;\;\;$ $\left[\text{Note: } a^{\log_a x} = x\right]$
$= \left[\left(\dfrac{1}{2}\right)^{\frac{-1}{2}} - \sqrt{3}\right]^2 + 2 \sqrt{6}$
$= \left[\left(\dfrac{1}{2^{\frac{1}{2}}}\right)^{-1} - \sqrt{3}\right]^2 + 2 \sqrt{6}$
$= \left[2^{\frac{1}{2}} - \sqrt{3}\right]^2 + 2 \sqrt{6}$
$= \left[\sqrt{2} - \sqrt{3}\right]^2 + 2 \sqrt{6}$
$= 2 + 3 - 2 \sqrt{6} + 2 \sqrt{6}$
$= 5$