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.