Let's analyze and simplify the given expression step by step.

The expression is:

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

Step 1: Simplify the first part inside the parentheses:

$\frac{x y^{-3}}{2 z^{-3}}$ This can be rewritten as: $\frac{x}{2} \cdot \frac{y^{-3}}{z^{-3}} = \frac{x}{2} \cdot \frac{z^3}{y^3} = \frac{x z^3}{2 y^3}$

Step 2: Apply the exponent to the simplified fraction:

$\left( \frac{x z^3}{2 y^3} \right)^2 = \frac{x^2 z^6}{4 y^6}$

Step 3: Multiply by the second part:

$\frac{x^2 z^6}{4 y^6} \cdot \frac{x^2}{z}$ This simplifies to: $\frac{x^2 z^6}{4 y^6} \cdot \frac{x^2}{z} = \frac{x^4 z^5}{4 y^6}$

Final Answer:

$\frac{x^4 z^5}{4 y^6}$

This is the simplified expression with only positive exponents.

Would you like a detailed breakdown of any step or further clarification?

Follow-up Questions:

1. How do we handle negative exponents in fractions?
2. Why does $z^{-3}$ become $z^3$ when moved from the denominator?
3. How does exponentiation affect each component of a fraction?
4. Can you identify other ways to simplify expressions with multiple variables?
5. What rules apply when multiplying variables with exponents?

Tip: Always remember that negative exponents indicate the reciprocal of the base raised to the positive exponent.