The value of $(\sqrt{x})^{\text{4log}_{x} \space a}$ is :

1. a

2. ax

3. $\dfrac{1}{2}ax$

4. a²

Given,

$\Rightarrow (\sqrt{x})^{\text{4log}${x} \space a} \\[1em] \Rightarrow (x^{\dfrac{1}{2}})^{\text{4log}{x} \space a} \\[1em] \Rightarrow (x)^{\dfrac{1}{2} \times \text{4log}{x} \space a} \\[1em] \Rightarrow (x)^{\text{2log}{x} \space a} \\[1em] \Rightarrow (x)^{\text{log}_{x} \space a^2} \\[1em] \Rightarrow a^2.

Hence, Option 4 is the correct option.