https://file.aiquickdraw.com/mathbot/file/1728832962683hyt3m4xx.jpg

The image displays the expression that needs to be simplified:

\[
\left( \frac{9x^{-\frac{2}{3}} y^{-\frac{1}{3}}}{xy^3} \right)^{-3}
\]

### Simplification process:
1. Start by simplifying the expression inside the parentheses.
   \[
   \frac{9x^{-\frac{2}{3}} y^{-\frac{1}{3}}}{xy^3}
   \]
   Break it into separate terms for \(x\) and \(y\):
   \[
   9 \cdot \frac{x^{-\frac{2}{3}}}{x} \cdot \frac{y^{-\frac{1}{3}}}{y^3}
   \]

2. For the \(x\)-terms:
   \[
   \frac{x^{-\frac{2}{3}}}{x} = x^{-\frac{2}{3} - 1} = x^{-\frac{5}{3}}
   \]

3. For the \(y\)-terms:
   \[
   \frac{y^{-\frac{1}{3}}}{y^3} = y^{-\frac{1}{3} - 3} = y^{-\frac{10}{3}}
   \]

4. The expression simplifies to:
   \[
   9x^{-\frac{5}{3}} y^{-\frac{10}{3}}
   \]

5. Now apply the exponent \(-3\) to the entire expression:
   \[
   \left( 9x^{-\frac{5}{3}} y^{-\frac{10}{3}} \right)^{-3}
   \]

6. Apply the power of \(-3\) to each term:
   \[
   9^{-3} \cdot x^{-\frac{5}{3} \cdot (-3)} \cdot y^{-\frac{10}{3} \cdot (-3)}
   \]
   This gives:
   \[
   \frac{1}{9^3} \cdot x^5 \cdot y^{10}
   \]

7. Calculate \(9^3 = 729\), so the final expression is:
   \[
   \frac{x^5 y^{10}}{729}
   \]

### Final answer:
\[
\frac{x^5 y^{10}}{729}
\]

Would you like any further clarification?

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Here are 5 related questions you might find useful:
1. How do you simplify more complex expressions with fractional exponents?
2. Can you explain the properties of exponents used here in more detail?
3. How would the solution change if a different power was applied outside the parentheses?
4. What is the significance of negative exponents in simplification?
5. Can you simplify an expression with variables in both the numerator and denominator?

**Tip:** Always apply exponent rules carefully, especially when dealing with fractions and negative exponents.

Simplify the expression (9x^(-2/3) * y^(-1/3) / (x * y^3))^(-3).

Learn how to simplify the expression (9x^(-2/3) * y^(-1/3) / (x * y^3))^(-3) step-by-step using exponent rules. This detailed solution covers handling fractional and negative exponents and applying power rules. Suitable for students in Grades 9-11.

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