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The given expression is:

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

### Step-by-Step Simplification:

1. **First, simplify the expression inside the parentheses**:
   - The first term is \( u z^2 \).
   - The second term is \( (-2x^4 u^3 z)^{-2} \).

2. **Apply the exponent of \(-2\) to the second term**:
   The term \( (-2x^4 u^3 z)^{-2} \) can be simplified by applying the exponent to each factor inside:
   \[
   (-2)^{-2} = \frac{1}{4}, \quad (x^4)^{-2} = x^{-8}, \quad (u^3)^{-2} = u^{-6}, \quad (z^{-1})^{-2} = z^{-2}
   \]
   So the second term becomes:
   \[
   \frac{1}{4} x^{-8} u^{-6} z^{-2}
   \]

3. **Multiply the two expressions together**:
   Now, multiply the first term \( u z^2 \) by the simplified second term:
   \[
   u z^2 \cdot \frac{1}{4} x^{-8} u^{-6} z^{-2}
   \]

   Multiply the corresponding variables:
   - For \( u \): \( u \cdot u^{-6} = u^{-5} \)
   - For \( z \): \( z^2 \cdot z^{-2} = z^0 = 1 \)
   - The \( x^{-8} \) and the constant \( \frac{1}{4} \) remain.

   Therefore, the result is:
   \[
   \frac{1}{4} x^{-8} u^{-5}
   \]

4. **Express the final answer using positive exponents**:
   \[
   \frac{x^8}{4 u^5}
   \]

### Final Answer:
\[
\frac{x^8}{4 u^5}
\]

Would you like more details or explanations on any part of the process?

### Related Questions:
1. How do you apply the exponent rule when multiplying powers with the same base?
2. What are the steps to simplify negative exponents?
3. How do you handle distributing an exponent over a product inside parentheses?
4. Can you explain the rules for multiplying variables with exponents?
5. How do you convert an expression with negative exponents to positive exponents?

### Tip:
When applying negative exponents, remember they represent the reciprocal of the base raised to the corresponding positive exponent.

(uz^2)(-2x^4 u^3 z)^{-2}. Write your answer using only positive exponents.

Learn how to simplify the expression (uz^2)(-2x^4 u^3 z)^{-2} step by step using exponent rules, including multiplication of powers and handling negative exponents. Suitable for students in Grades 9-11.

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