The expression you've shared is as follows:

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

Let’s simplify this expression step by step.

Step 1: Simplify each part separately

1. Simplify $(x^{-3} y^2 z^3)^2$:

   • Apply the power to each component inside the parentheses.
   • $x^{-3} \rightarrow x^{-6}$
   • $y^2 \rightarrow y^4$
   • $z^3 \rightarrow z^6$
   • So, $(x^{-3} y^2 z^3)^2 = x^{-6} y^4 z^6$.

2. Simplify $(xyz^2)^{-1}$:

   • Apply the $-1$ power to each term inside.
   • $x \rightarrow x^{-1}$
   • $y \rightarrow y^{-1}$
   • $z^2 \rightarrow z^{-2}$
   • So, $(xyz^2)^{-1} = x^{-1} y^{-1} z^{-2}$.

Step 2: Combine the numerator

Now, the numerator becomes: $x^{-6} y^4 z^6 \cdot x^{-1} y^{-1} z^{-2} = x^{-7} y^3 z^4$

Step 3: Simplify the denominator

The denominator is: $x^2 y^{-2} z^{-1}$

Step 4: Divide the numerator by the denominator

Now we perform the division by subtracting the exponents of like terms:

1. For $x$: $-7 - 2 = -9$
2. For $y$: $3 - (-2) = 3 + 2 = 5$
3. For $z$: $4 - (-1) = 4 + 1 = 5$

So the simplified form of the expression is: $x^{-9} y^5 z^5$

Final Answer

$\frac{(x^{-3} y^2 z^3)^2 (xyz^2)^{-1}}{x^2 y^{-2} z^{-1}} = x^{-9} y^5 z^5$

Would you like further clarification on any of the steps?

Here are some related questions that may deepen your understanding:

1. How do you handle negative exponents in expressions?
2. What are the rules for multiplying terms with exponents?
3. How do you simplify expressions with both negative and positive exponents?
4. What happens when you apply a negative exponent to an entire term?
5. How would this expression change if we added terms instead of multiplying?

Tip: Remember that dividing terms with the same base means subtracting the exponents (e.g., $x^a / x^b = x^{a-b}$).