Given,

${\left(2.0001\right)}^3\times {\left(1.999\right)}^{-2}÷{\left(3.999\right)}^{-4}$

It can also be approximated

$$\begin{gathered} =\begin{array}{l}{\left(\text{2}\right)}^{\text{3}}\times {\left(\text{2}\right)}^{\text{-2}}÷{\left(\text{4}\right)}^{-4}\end{array} \\ \begin{array}{l}{\text{=(2)}}^{\text{3}}\times \left(2^{-2}\right)÷{\left(4\right)}^{-4}\end{array} \\ \begin{array}{l}{\text{=(2)}}^{\text{3}}\times {\left(2\right)}^{-2+8}\\ \end{array} \\ {\text{=2}}^{\text{3}}\times 2^{8-2}=2^3\times 2^6=2^9=512 \end{gathered}$$

Hence, Option E is correct.