https://file.aiquickdraw.com/mathbot/file/1727642703316rd4q4j61.jpg

The expression shown in the image is:

\[
\left( \frac{z^3}{8} \right)^{-3}
\]

### Steps to Simplify:
1. Apply the exponent rule: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)
   \[
   \left( \frac{z^3}{8} \right)^{-3} = \frac{(z^3)^{-3}}{8^{-3}}
   \]

2. Simplify each part:
   - \( (z^3)^{-3} = z^{3 \times -3} = z^{-9} \)
   - \( 8^{-3} = \frac{1}{8^3} = \frac{1}{512} \)

3. Therefore, the expression becomes:
   \[
   \frac{z^{-9}}{\frac{1}{512}} = z^{-9} \times 512
   \]

4. Simplify the final expression:
   \[
   512z^{-9} = \frac{512}{z^9}
   \]

### Final Answer:
\[
\frac{512}{z^9}
\]

Let me know if you'd like any further explanation!

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**Related Questions:**
1. What is the general rule for simplifying negative exponents?
2. How do you handle fractions raised to a negative power?
3. Can the same simplification method be used for variables with different exponents?
4. What happens if the denominator is a variable instead of a constant?
5. How does this rule apply to more complex fractions?

**Tip:** Always simplify powers of variables and constants separately, then combine them.

Learn how to simplify the expression (z^3 / 8)^(-3) step by step using exponent rules. This problem covers key algebraic concepts, including the laws of exponents and fractional powers. Suitable for students in Grades 8-10.

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