Evaluate the following :

$\Big(\dfrac{16}{81}\Big)^{-\dfrac{3}{4}} \times \Big(\dfrac{49}{9}\Big)^{\dfrac{3}{2}} ÷ \Big(\dfrac{343}{216}\Big)^{\dfrac{2}{3}}$

Given,

$\Big(\dfrac{16}{81}\Big)^{-\dfrac{3}{4}} \times \Big(\dfrac{49}{9}\Big)^{\dfrac{3}{2}} ÷ \Big(\dfrac{343}{216}\Big)^{\dfrac{2}{3}}$

Simplifying the expression :

$\Rightarrow \Big(\dfrac{16}{81}\Big)^{-\dfrac{3}{4}} \times \Big(\dfrac{49}{9}\Big)^{\dfrac{3}{2}} ÷ \Big(\dfrac{343}{216}\Big)^{\dfrac{2}{3}} \\[1em] \Rightarrow \Big(\dfrac{81}{16}\Big)^{\dfrac{3}{4}} \times \Big[\Big(\dfrac{7}{3}\Big)^2\Big]^{\dfrac{3}{2}} ÷ \Big[\Big(\dfrac{7}{6}\Big)^3\Big]^{\dfrac{2}{3}} \\[1em] \Rightarrow \Big[\Big(\dfrac{3}{2}\Big)^4\Big]^{\dfrac{3}{4}} \times \Big[\Big(\dfrac{7}{3}\Big)^2\Big]^{\dfrac{3}{2}} ÷ \Big[\Big(\dfrac{7}{6}\Big)^3\Big]^{\dfrac{2}{3}} \\[1em] \Rightarrow \Big(\dfrac{3}{2}\Big)^3 \times \Big(\dfrac{7}{3}\Big)^3 ÷ \Big(\dfrac{7}{6}\Big)^2 \\[1em] \Rightarrow \Big(\dfrac{27}{8}\Big) \times \Big[\Big(\dfrac{343}{27}\Big) ÷ \Big(\dfrac{49}{36}\Big)\Big] \\[1em] \Rightarrow \Big(\dfrac{27}{8}\Big) \times \Big[\Big(\dfrac{343}{27}\Big) \times \Big(\dfrac{36}{49}\Big)\Big] \\[1em] \Rightarrow \Big(\dfrac{27}{8}\Big) \times \Big(\dfrac{7}{3}\Big) \times 4 \\[1em] \Rightarrow \Big(\dfrac{9}{2}\Big) \times 7 \\[1em] \Rightarrow \dfrac{63}{2} = 31\dfrac{1}{2}.$

Hence, $\Big(\dfrac{16}{81}\Big)^{-\dfrac{3}{4}} \times \Big(\dfrac{49}{9}\Big)^{\dfrac{3}{2}} ÷ \Big(\dfrac{343}{216}\Big)^{\dfrac{2}{3}} = 31\dfrac{1}{2}$.