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^{-6} \cdot y^4 \cdot z^{-1}$$
2. Simplify the second term: $\left( x^{-2} y z^{\frac{2}{3}} \right)^{-3}$ Applying the exponent -3 to each factor: = x^6 \cdot y^{-3} \cdot z^{-2}$$
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^0 \cdot y^1 \cdot z^{-3} = y \cdot z^{-3}$$

Thus, the simplified expression is $yz^{-3}$.

The correct answer is:

D. $yz^{-3}$

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!