The value of $\Big(5\dfrac{1}{16}\Big)^{-\dfrac{3}{4}}$ is

1. $\dfrac{4}{9}$

2. $\dfrac{9}{4}$

3. $\dfrac{27}{8}$

4. $\dfrac{8}{27}$

Given,

$\Rightarrow \Big(5\dfrac{1}{16}\Big)^{-\dfrac{3}{4}} = \Big(\dfrac{81}{16}\Big)^{-\dfrac{3}{4}} \\[1em] = \Big(\dfrac{16}{81}\Big)^{\dfrac{3}{4}} \\[1em] = \Big(\dfrac{2^4}{3^4}\Big)^{\dfrac{3}{4}} = \dfrac{2^{4 \times \frac{3}{4}}}{3^{4 \times \dfrac{3}{4}}} \\[1em] = \dfrac{2^3}{3^3} = \dfrac{8}{27}.$

Hence, Option 4 is the correct option.