Evaluate :

$\Big[\Big(-\dfrac{2}{3}\Big)^{-2}\Big]^3 \times \Big(\dfrac{1}{3}\Big)^{-4} \times 3^{-1} \times \dfrac{1}{6}$

Simplify the expression :

$\Rightarrow \Big[\Big(-\dfrac{2}{3}\Big)^{-2}\Big]^3 \times \Big(\dfrac{1}{3}\Big)^{-4} \times 3^{-1} \times \dfrac{1}{6} \\[1em] = \Big(-\dfrac{2}{3}\Big)^{-6} \times (3^{-1})^{-4} \times \dfrac{1}{3} \times \dfrac{1}{6} \\[1em] = \Big(\dfrac{3}{2}\Big)^6 \times 3^4 \times \dfrac{1}{18} \\[1em] = \dfrac{3^6 \times 3^4}{2^6 \times 18} \\[1em] = \dfrac{3^{6 + 4}}{2^6 \times (2 \times 3 \times 3)} \\[1em] = \dfrac{3^{10}}{2^{6 + 1} \times 3^2} \\[1em] = \dfrac{3^{10 - 2}}{2^7} \\[1em] = \dfrac{3^8}{2^7} \\[1em] = 3^8 ÷ 2^7.$

Hence, $\Big[\Big(-\dfrac{2}{3}\Big)^{-2}\Big]^3 \times \Big(\dfrac{1}{3}\Big)^{-4} \times 3^{-1} \times \dfrac{1}{6} = 3^8 ÷ 2^7.$