$\begin{align} & 4{{\left( {{3}^{2k-1}} \right)}^{3}}=a\Rightarrow 4\times {{3}^{\left( 6k-3 \right)}}=a \\ & 2{{\left( {{3}^{3-2k}} \right)}^{-3}}=b\Rightarrow 2\times \left( {{3}^{-9+6k}} \right)=b \\ & \frac{a}{b}=\frac{4\times {{3}^{\left( 6k-3 \right)}}}{2\times {{3}^{\left( -9+6k \right)}}}=2\times {{3}^{6k-3-\left( -9+6k \right)}}=2\times {{3}^{6k-3+9-6k}}\Rightarrow \frac{a}{b}=2\times {{3}^{6}}=2\times 729=1458 \\ \end{align}$