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The given mathematical expression is:

\[
\left( \frac{w^{-2}x^4z^6}{w^3x^2z^8} \right)^{-3}
\]

Let's simplify it step by step:

1. **Simplify the expression inside the parentheses:**

   - For the variable \(w\):
     \[
     \frac{w^{-2}}{w^3} = w^{-2-3} = w^{-5}
     \]

   - For the variable \(x\):
     \[
     \frac{x^4}{x^2} = x^{4-2} = x^2
     \]

   - For the variable \(z\):
     \[
     \frac{z^6}{z^8} = z^{6-8} = z^{-2}
     \]

   The simplified expression becomes:
   \[
   \left( w^{-5}x^2z^{-2} \right)^{-3}
   \]

2. **Apply the power of -3 to each term:**

   - For \(w^{-5}\):
     \[
     (w^{-5})^{-3} = w^{15}
     \]

   - For \(x^2\):
     \[
     (x^2)^{-3} = x^{-6}
     \]

   - For \(z^{-2}\):
     \[
     (z^{-2})^{-3} = z^6
     \]

   Therefore, the final simplified expression is:
   \[
   w^{15}x^{-6}z^6
   \]

   This can also be written as:
   \[
   \frac{w^{15}z^6}{x^6}
   \]

Would you like more details or have any questions?

Here are some related questions to explore:
1. How do you simplify expressions with negative exponents?
2. What are the properties of exponents used in this problem?
3. How can you apply the power of a quotient rule to simplify more complex expressions?
4. What is the general rule for raising a power to another power?
5. How can you convert expressions with negative exponents into fractions?

**Tip:** When simplifying expressions with exponents, remember to handle each variable independently according to exponent rules before combining results.

(w^-2 * x^4 * z^6) / (w^3 * x^2 * z^8) all raised to the power of -3

Learn how to simplify the expression (w^-2 * x^4 * z^6) / (w^3 * x^2 * z^8) raised to the power of -3. This problem demonstrates exponent rules, including the power of a quotient rule and handling negative exponents. Suitable for Grades 9-12 students.

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