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The image shows a math problem asking to simplify an expression under the assumption that \(x\) and \(y\) are variables. The given expression is:

\[
\left( \frac{x^9}{27y^{-12}} \right)^{\frac{2}{3}}
\]

### Steps to Simplify:

1. **Rewrite the base expression**:
   \[
   \frac{x^9}{27y^{-12}} = x^9 \cdot \frac{1}{27} \cdot y^{12}
   \]

   Here, \(y^{-12}\) becomes \(y^{12}\) in the numerator.

2. **Simplify constants**:
   Keep the \(x^9\), \(\frac{1}{27}\), and \(y^{12}\) together:
   \[
   \frac{x^9 \cdot y^{12}}{27}
   \]

3. **Apply the exponent \(\frac{2}{3}\)**:
   Distribute the exponent \(\frac{2}{3}\) to all parts of the expression:
   \[
   \left( \frac{x^9 \cdot y^{12}}{27} \right)^{\frac{2}{3}} = \frac{x^{9 \cdot \frac{2}{3}} \cdot y^{12 \cdot \frac{2}{3}}}{27^{\frac{2}{3}}}
   \]

4. **Simplify the exponents**:
   - \(x^{9 \cdot \frac{2}{3}} = x^6\)
   - \(y^{12 \cdot \frac{2}{3}} = y^8\)
   - \(27^{\frac{2}{3}}\) is the cube root of \(27\) squared: \(\sqrt[3]{27^2} = 9\)

   So the expression becomes:
   \[
   \frac{x^6 \cdot y^8}{9}
   \]

Thus, the simplified expression is:

\[
\boxed{\frac{x^6 y^8}{9}}
\]

Would you like any further details or explanations on this?

Here are 5 related questions you might find interesting:
1. How do you simplify expressions involving negative exponents?
2. What is the general method for applying fractional exponents?
3. How do you handle cube roots of larger numbers?
4. What are the rules for multiplying powers with the same base?
5. How does raising a fraction to a power affect both the numerator and the denominator?

**Tip**: When dealing with fractional exponents, remember that the numerator represents a power, and the denominator represents a root.

Simplify the expression (x^9 / 27y^(-12))^(2/3), assuming x and y are variables.

Learn how to simplify the expression (x^9 / 27y^(-12))^(2/3) step-by-step. This problem covers key algebraic concepts, including exponents, negative exponents, and fractional exponents. Suitable for students in Grades 9-11.

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