On simplifying $\Big[5\Big(8^\dfrac{1}{3} + 27^\dfrac{1}{3}\Big)^{3}\Big]^{\dfrac{1}{4}}$, we get:

1. 5

2. $\sqrt{5}$

3. 10

4. 1

Given,

$\Big[5\Big(8^\dfrac{1}{3} + 27^\dfrac{1}{3}\Big)^{3}\Big]^{\dfrac{1}{4}}$

Now simplifying:

$\Rightarrow \Big[5\Big((2^3)^{\dfrac{1}{3}} + (3^3)^{\dfrac{1}{3}}\Big)^{3}\Big]^{\dfrac{1}{4}} \\[1em] \Rightarrow \Big[5(2 + 3)^{3}\Big]^{\dfrac{1}{4}} \\[1em] \Rightarrow \Big[5(5)^{3}\Big]^{\dfrac{1}{4}} \\[1em] \Rightarrow \Big[(5)^{4}\Big]^{\dfrac{1}{4}} \\[1em] \Rightarrow 5^{4 \times \dfrac{1}{4}} \\[1em] \Rightarrow 5.$

Hence, option 1 is the correct option.