The expression you've provided is quite complex, so let’s break it down step-by-step.

The expression is: [ \left( \sqrt{\frac{(1 - n) \cdot \sqrt[3]{1 + n} \cdot \sqrt[3]{\frac{3n^2}{4 - 8n + 4n^2}}}{n}} \right)^{-1} \bigg/ \left( \sqrt[3]{\frac{3n \sqrt{n}}{2 \sqrt{1 - n^2}}} \right)^{-1} ]

To solve this, let’s follow these steps:

1. Simplify each part inside the square roots and cube roots.
2. Rewrite negative exponents by reciprocating the expressions.
3. Evaluate any further simplifications possible by factoring or reducing the terms.

I'll start by evaluating each section. Would you like a detailed step-by-step simplification, or are you looking for a specific final form?