Let's analyze and solve the expression in the image. The expression is:

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

To simplify this, we can handle each variable's exponent separately using the rules of exponents. Specifically, for division, we can subtract the exponents of like bases in the numerator and denominator.

Step-by-Step Solution:

1. Separate Variables by Base: $\frac{x^{-3}}{x^2} \cdot \frac{y^{-1}}{y^{-2}} \cdot \frac{z^2}{z^{-1}}$

2. Apply the Quotient Rule for Exponents:

   • For $x$: $x^{-3 - 2} = x^{-5}$
   • For $y$: $y^{-1 - (-2)} = y^{1}$
   • For $z$: $z^{2 - (-1)} = z^{3}$

   So, the expression simplifies to: $x^{-5} y^{1} z^{3}$

3. Rewrite Negative Exponents as Fractions:

   • $x^{-5} = \frac{1}{x^5}$

   Thus, the expression becomes: $\frac{y z^3}{x^5}$

Final Answer:

$\frac{y z^3}{x^5}$

Would you like a detailed explanation of each exponent rule used, or do you have any further questions?

Here are some related questions you might find interesting:

1. How do negative exponents affect division in algebraic expressions?
2. What are the steps to simplify expressions with multiple variables?
3. How do we handle expressions that involve negative and positive exponents?
4. How can we simplify expressions involving multiple variables and fractions?
5. How would the solution change if the exponents were different?

Tip: Remember, when dividing like bases, subtract the exponents, and rewrite negative exponents as fractions for simplification.