The given expression is:

$(a - b)^3 \cdot \left( \frac{a + b}{b - a} \right)^{-2}$

Step 1: Simplify the second part

The expression $\frac{a + b}{b - a}$ can be rewritten as:

$\frac{a + b}{b - a} = - \frac{a + b}{a - b}$

Thus, the second part becomes:

$\left( \frac{a + b}{b - a} \right)^{-2} = \left( - \frac{a + b}{a - b} \right)^{-2} = \frac{(a - b)^2}{(a + b)^2}$

Step 2: Multiply the two terms

Now, the entire expression becomes:

$(a - b)^3 \cdot \frac{(a - b)^2}{(a + b)^2}$

Step 3: Combine powers of $(a - b)$

Using the property of exponents, $(a - b)^3 \cdot (a - b)^2 = (a - b)^5$, the expression simplifies to:

$\frac{(a - b)^5}{(a + b)^2}$

Thus, the simplified expression is:

$\boxed{\frac{(a - b)^5}{(a + b)^2}}$

Would you like any further details or have any questions about the steps?

Here are some related questions to practice:

1. How do you simplify powers when multiplying terms with the same base?
2. What is the general rule for negative exponents?
3. How do you handle a fraction raised to a negative power?
4. Can you provide another example of combining powers with the same base?
5. What happens when simplifying expressions involving both powers and negative signs?

Tip: Always pay attention to signs when working with expressions like $(a - b)$ and $(b - a)$ since they are negatives of each other!