$T_{r+1}=18 C_{r}\left(x^{\frac{1}{3}}\right)^{18-r}\left(\frac{1}{2 x^{\frac{1}{3}}}\right)^{r}$

$=^{18} C_{r} x^{6-\frac{2 r}{3}} \frac{1}{2^{r}}$

$\left\{ \begin{gathered}
  6 – \frac{{2r}}{3} =  – 2 \Rightarrow r = 12 \hfill \\
  \& \,6 – \frac{{2r}}{3} =  – 4 \Rightarrow r = 15 \hfill \\ 
\end{gathered}  \right\}$

$\Rightarrow \quad \frac{\text { coefficient of } x^{-2}}{\text { coefficient of } x^{-4}}=\frac{^{18} C_{12} \frac{1}{2^{12}}}{^{18} C_{15} \frac{1}{2^{15}}}=182$