The expression in the image is:

$\left( 4a^{-3}b^2 \right)^3$

Let’s simplify this step-by-step:

Step 1: Apply the power rule

When an entire expression is raised to a power, distribute the power to each term inside. Using the property $(x^m)^n = x^{m \cdot n}$, we have:

$\left( 4a^{-3}b^2 \right)^3 = 4^3 \cdot (a^{-3})^3 \cdot (b^2)^3$

Step 2: Simplify each component

1. $4^3 = 64$
2. $(a^{-3})^3 = a^{-9}$ (multiply the exponents: $-3 \cdot 3 = -9$)
3. $(b^2)^3 = b^6$ (multiply the exponents: $2 \cdot 3 = 6$)

Step 3: Combine the results

$64 \cdot a^{-9} \cdot b^6$

Step 4: Write with positive exponents, if necessary

If needed, rewrite $a^{-9}$ as $\frac{1}{a^9}$. Thus:

$\frac{64b^6}{a^9}$

Final Answer:

$\boxed{64a^{-9}b^6 \text{ or } \frac{64b^6}{a^9}}$

Let me know if you need more details or explanations!

Here are 5 questions you might find helpful:

1. What is the general rule for handling negative exponents?
2. How would this change if the power was $-3$ instead of $3$?
3. Can you apply this concept to simplify $(2x^{-4}y)^5$?
4. Why is it important to distribute the exponent to every term inside the parentheses?
5. How does writing with positive exponents affect the clarity of an expression?

Tip: Always check if the question requires leaving exponents negative or converting them to fractions!