The given mathematical expression is:

$\sqrt{\frac{(1 - x)^3}{1 + x}}$

Simplification:

1. The square root applies to the entire fraction: $\sqrt{\frac{(1 - x)^3}{1 + x}} = \frac{\sqrt{(1 - x)^3}}{\sqrt{1 + x}}$

2. The numerator $\sqrt{(1 - x)^3}$ can be rewritten as: $\sqrt{(1 - x)^3} = (1 - x)^{\frac{3}{2}}$

3. The denominator remains: $\sqrt{1 + x} = (1 + x)^{\frac{1}{2}}$

4. So the expression becomes: $\frac{(1 - x)^{\frac{3}{2}}}{(1 + x)^{\frac{1}{2}}}$

This is the simplified form of the given expression.

Would you like further explanations or examples? Here are some related questions:

1. How do you rationalize a fraction with square roots?
2. What are the conditions for this expression to be real?
3. How can we expand the numerator using binomial expansion?
4. How does this function behave graphically?
5. How can we differentiate or integrate this function?

Tip: When dealing with square roots and exponents, always check domain restrictions to ensure valid calculations!