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The problem requires simplifying the expression \( \left( \frac{1}{z^9} \right)^3 \).

Here's the solution:

1. Start with the expression:
   \[
   \left( \frac{1}{z^9} \right)^3
   \]

2. Apply the power of a power rule, which states \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \). In this case, we get:
   \[
   \frac{1^3}{(z^9)^3}
   \]

3. Simplify \(1^3\) and \( (z^9)^3 \):
   - \(1^3 = 1\),
   - \((z^9)^3 = z^{9 \times 3} = z^{27}\).

4. The expression simplifies to:
   \[
   \frac{1}{z^{27}}
   \]

The correct answer is:
\[
\frac{1}{z^{27}}
\]

So, the answer is the third option.

Would you like more details on this problem, or have any other questions?

Here are some related questions:
1. How do you apply the power of a power rule in other expressions?
2. What is the difference between multiplying and raising powers?
3. How would you simplify \(\left( \frac{1}{x^4} \right)^5\)?
4. How can you rewrite expressions with negative exponents?
5. Why is \(1\) raised to any power always \(1\)?

**Tip:** Remember, when raising a power to another power, multiply the exponents.

Which expression is equivalent to (1/z^9)^3?

Learn how to simplify the expression (1/z^9)^3 by applying the power of a power rule. This problem demonstrates exponent rules, including simplifying powers and handling fractions in exponents. Suitable for students in Grades 9-10.

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