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The problem shown asks to simplify the following expression:

\[
\left( \frac{x^{-3} y^{5} z^{4}}{y^{-2} z^{-4}} \right)^2
\]

### Step-by-step Solution:

1. **Simplify inside the parentheses:**

   \[
   \frac{x^{-3} y^{5} z^{4}}{y^{-2} z^{-4}}
   \]

   For the variables with the same base (like \( y \) and \( z \)), use the rule for dividing exponents: 
   
   \[
   \frac{a^m}{a^n} = a^{m - n}
   \]
   
   So:
   \[
   y^{5 - (-2)} = y^{5 + 2} = y^{7}
   \]
   
   \[
   z^{4 - (-4)} = z^{4 + 4} = z^{8}
   \]
   
   Therefore, the expression becomes:
   \[
   x^{-3} y^{7} z^{8}
   \]

2. **Apply the exponent outside the parentheses:**

   Now we need to apply the exponent \( 2 \) to each term inside the parentheses. Use the power rule \( (a^m)^n = a^{m \cdot n} \):
   
   \[
   (x^{-3})^2 = x^{-6}
   \]
   
   \[
   (y^{7})^2 = y^{14}
   \]
   
   \[
   (z^{8})^2 = z^{16}
   \]

   So the expression becomes:
   \[
   x^{-6} y^{14} z^{16}
   \]

3. **Final Answer:**

   To express the answer with positive exponents, move \( x^{-6} \) to the denominator:
   
   \[
   \frac{y^{14} z^{16}}{x^{6}}
   \]

Thus, the simplified expression is:

\[
\frac{y^{14} z^{16}}{x^6}
\]

---

Would you like further details on any step or have any questions?

### Related Questions:
1. How do you simplify expressions with negative exponents?
2. What is the rule for multiplying powers with the same base?
3. How can you expand an expression raised to a power?
4. What are some common mistakes when simplifying exponents?
5. How do you convert negative exponents to positive ones?

### Tip:
Always simplify expressions inside parentheses before applying outer exponents.

Simplify the expression ((x^(-3) * y^5 * z^4) / (y^(-2) * z^(-4)))^2

Learn how to simplify the expression ((x^(-3) * y^5 * z^4) / (y^(-2) * z^(-4)))^2 using exponent laws. This problem involves dividing and multiplying powers with negative exponents and simplifying the result.

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