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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.

Simplify the following expression: ((x y^(-3))/(2 z^(-3)))^2 * (x^2/z). Write your answer using only positive exponents.

Simplify an algebraic expression involving negative exponents: ((x y^(-3))/(2 z^(-3)))^2 * (x^2/z). This problem demonstrates how to handle negative exponents and apply exponentiation to fractions step-by-step. Suitable for Grades 8-10.

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