The expression in the image is:

$\frac{(-4)^3}{[(-2)^2 \cdot (-2)^3]}$

Let's break it down and solve it step by step:

Step 1: Simplify $(-4)^3$

$(-4)^3 = -4 \times -4 \times -4 = -64$

Step 2: Simplify the denominator

• $(-2)^2 = (-2) \times (-2) = 4$
• $(-2)^3 = (-2) \times (-2) \times (-2) = -8$

Now, multiply the two parts of the denominator: $4 \cdot (-8) = -32$

Step 3: Combine the results

Now, the expression becomes: $\frac{-64}{-32}$

Simplifying this gives: $\frac{-64}{-32} = 2$

Final Answer:

$2$

Would you like more details or have any questions?

Here are some related questions:

1. How do you simplify other powers of negative numbers?
2. What are the rules for multiplying powers of the same base?
3. How do you handle exponents when they are fractions?
4. What is the difference between $(-a)^n$ and $-a^n$?
5. How do you simplify complex fractions involving powers?

Tip: Always be careful with negative signs inside and outside exponents—they can drastically change the result.