https://file.aiquickdraw.com/mathbot/file/1726549159330vpjc5fy9.jpg

The expression displayed is:

\[
\left(\frac{x^{-9}}{y^{-6}}\right)^{-\frac{1}{3}}
\]

To simplify this expression and ensure all exponents are positive, follow these steps:

### Step 1: Apply the Power of a Quotient Rule
We can distribute the outer exponent \(-\frac{1}{3}\) to both the numerator and denominator:
\[
\left(\frac{x^{-9}}{y^{-6}}\right)^{-\frac{1}{3}} = \frac{\left(x^{-9}\right)^{-\frac{1}{3}}}{\left(y^{-6}\right)^{-\frac{1}{3}}}
\]

### Step 2: Multiply the Exponents
Apply the property of exponents \((a^m)^n = a^{m \cdot n}\):
\[
\left(x^{-9}\right)^{-\frac{1}{3}} = x^{3}, \quad \left(y^{-6}\right)^{-\frac{1}{3}} = y^{2}
\]

### Step 3: Write the Final Expression
The simplified expression becomes:
\[
\frac{x^3}{y^2}
\]

Thus, the simplified expression with only positive exponents is:
\[
\boxed{\frac{x^3}{y^2}}
\]

Would you like further details or have any questions?

Here are five related questions:
1. How do you apply the rules of exponents when there are negative exponents?
2. How would you handle an exponent that is a fraction?
3. What is the process for simplifying a product of powers with different bases?
4. Can exponent rules be applied if variables are not positive?
5. What is the significance of writing expressions with only positive exponents?

**Tip:** Always apply exponent rules carefully when working with negative and fractional exponents to avoid mistakes.

Simplify the expression (x^-9 / y^-6)^(-1/3), leaving your answer with only positive exponents.

Simplify the expression (x^-9 / y^-6)^(-1/3) using exponent rules, ensuring all exponents are positive. Follow step-by-step instructions applying the Power of a Quotient rule and Negative Exponent rule, suitable for students in Grades 9-12.

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