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The given mathematical expression is:

\[
\frac{8x^{-2}y^5z^2}{4x^{-3}y^3z^3}
\]

### Step 1: Simplify the coefficients
The numerical coefficients are \(\frac{8}{4} = 2\).

### Step 2: Simplify the variables
- For \(x\), we apply the rule for dividing exponents: \(\frac{x^a}{x^b} = x^{a-b}\). So:
  \[
  \frac{x^{-2}}{x^{-3}} = x^{-2 - (-3)} = x^{1}
  \]
  
- For \(y\), similarly:
  \[
  \frac{y^5}{y^3} = y^{5-3} = y^2
  \]
  
- For \(z\):
  \[
  \frac{z^2}{z^3} = z^{2-3} = z^{-1} = \frac{1}{z}
  \]

### Step 3: Combine the results
Now we combine the simplified terms:

\[
2x^1y^2 \cdot \frac{1}{z} = \frac{2xy^2}{z}
\]

### Final Answer:
The simplified form is \(\frac{2xy^2}{z}\), which corresponds to option **e**.

Would you like more details or have any questions? 

Here are 5 related questions to expand your understanding:
1. How would the expression change if the exponents were negative for both the numerator and denominator?
2. What are the rules for dividing variables with exponents?
3. How can you generalize this approach for any expression involving exponents and division?
4. How does the rule for negative exponents work when simplifying such fractions?
5. How would the expression look if there were no negative exponents?

**Tip:** Always remember that when dividing powers with the same base, subtract the exponents!

Bentuk sederhana dari \(\frac{8x^{-2}y^5z^2}{4x^{-3}y^3z^3}\) adalah?

Learn how to simplify the expression \(\frac{8x^{-2}y^5z^2}{4x^{-3}y^3z^3}\) using exponent rules. This problem demonstrates the laws of exponents, including how to subtract exponents when dividing terms. Suitable for students in Grades 9-11.

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