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The question asks to simplify the following expression:

\[
\left( x^3y^{-2}z^{\frac{1}{2}} \right)^{-2} \cdot \left( x^{-2} y z^{\frac{2}{3}} \right)^{-3}
\]

### Step-by-step simplification:

1. **Simplify the first term:**  
   \[
   \left( x^3y^{-2}z^{\frac{1}{2}} \right)^{-2}
   \]
   Applying the exponent `-2` to each factor:
   \[
   = x^{3 \cdot (-2)} \cdot y^{-2 \cdot (-2)} \cdot z^{\frac{1}{2} \cdot (-2)}
   = x^{-6} \cdot y^4 \cdot z^{-1}
   \]

2. **Simplify the second term:**
   \[
   \left( x^{-2} y z^{\frac{2}{3}} \right)^{-3}
   \]
   Applying the exponent `-3` to each factor:
   \[
   = x^{-2 \cdot (-3)} \cdot y^{1 \cdot (-3)} \cdot z^{\frac{2}{3} \cdot (-3)}
   = x^6 \cdot y^{-3} \cdot z^{-2}
   \]

3. **Multiply the simplified terms together:**
   Now, multiply the results of both terms:
   \[
   (x^{-6} \cdot y^4 \cdot z^{-1}) \cdot (x^6 \cdot y^{-3} \cdot z^{-2})
   \]
   Combine the powers of the same variables:
   \[
   x^{-6+6} \cdot y^{4+(-3)} \cdot z^{-1+(-2)}
   = x^0 \cdot y^1 \cdot z^{-3}
   = y \cdot z^{-3}
   \]

Thus, the simplified expression is \( yz^{-3} \).

### The correct answer is:
**D. \( yz^{-3} \)**

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Do you have any questions or would you like further details on the steps?

Here are 5 related questions to consider:
1. How do you apply negative exponents to simplify terms?
2. How do you combine exponents when multiplying like bases?
3. What happens when you raise a power to another power?
4. How do you handle fractional exponents in expressions?
5. Can you identify different forms of exponential simplifications?

**Tip:** When dealing with exponents, always apply the exponent to every factor inside the parentheses!

Bentuk sederhana dari (x^3 y^(-2) z^(1/2))^(-2) . (x^(-2) y z^(2/3))^(-3) adalah . . .

Learn how to simplify expressions involving exponents like (x^3 y^(-2) z^(1/2))^(-2) and (x^(-2) y z^(2/3))^(-3) step by step. Understand key algebraic concepts including the laws of exponents and exponentiation rules. Suitable for high school students.

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