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!