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Let's begin by simplifying the given expression step by step. The expression is:

\[
\left( \frac{3r^{-4} s^2}{2r^3 s} \right)^{-3} \left( \frac{6r^2 s^3}{8r s^5} \right)^2
\]

### Step 1: Simplify the terms inside each set of parentheses.

#### First expression:
\[
\frac{3r^{-4} s^2}{2r^3 s}
\]

- For the \(r\)-terms: \(r^{-4}\) divided by \(r^3\) gives \(r^{-4 - 3} = r^{-7}\).
- For the \(s\)-terms: \(s^2\) divided by \(s\) gives \(s^{2 - 1} = s^1 = s\).

So, the first expression simplifies to:
\[
\frac{3 r^{-7} s}{2}
\]

#### Second expression:
\[
\frac{6r^2 s^3}{8r s^5}
\]

- For the \(r\)-terms: \(r^2\) divided by \(r\) gives \(r^{2 - 1} = r\).
- For the \(s\)-terms: \(s^3\) divided by \(s^5\) gives \(s^{3 - 5} = s^{-2}\).

So, the second expression simplifies to:
\[
\frac{6 r s^{-2}}{8}
\]

Now, simplify the constants in both expressions.

#### First expression:
\[
\frac{3}{2} r^{-7} s
\]

#### Second expression:
\[
\frac{6}{8} = \frac{3}{4}, \quad \text{so it becomes:} \quad \frac{3}{4} r s^{-2}
\]

### Step 2: Apply the exponents.

Now apply the exponents \(-3\) and \(2\) to both expressions.

#### First expression raised to \(-3\):
\[
\left( \frac{3}{2} r^{-7} s \right)^{-3} = \frac{(2)^3}{(3)^3} r^{21} s^{-3} = \frac{8 r^{21} s^{-3}}{27}
\]

#### Second expression raised to \(2\):
\[
\left( \frac{3}{4} r s^{-2} \right)^2 = \frac{(3)^2}{(4)^2} r^2 s^{-4} = \frac{9 r^2 s^{-4}}{16}
\]

### Step 3: Multiply the two results.

Now, multiply the two simplified expressions:

\[
\frac{8 r^{21} s^{-3}}{27} \times \frac{9 r^2 s^{-4}}{16}
\]

- Multiply the constants: \(\frac{8 \times 9}{27 \times 16} = \frac{72}{432} = \frac{1}{6}\).
- Multiply the \(r\)-terms: \(r^{21} \times r^2 = r^{21 + 2} = r^{23}\).
- Multiply the \(s\)-terms: \(s^{-3} \times s^{-4} = s^{-3 - 4} = s^{-7}\).

Thus, the final expression simplifies to:
\[
\frac{r^{23} s^{-7}}{6}
\]

Or equivalently:
\[
\frac{r^{23}}{6 s^7}
\]

This is the fully simplified expression.

Would you like more details or have any questions?

### Related questions:
1. How do you simplify negative exponents when dividing terms with the same base?
2. What are the rules for raising a fraction to a negative exponent?
3. How does distributing exponents over a fraction work?
4. Can you explain how to multiply terms with exponents in detail?
5. What are the common mistakes when simplifying algebraic expressions involving exponents?

### Tip:
When working with exponents, always apply exponent rules carefully, especially when dealing with negative or fractional exponents.

Simplify the expression: \left( \frac{3r^{-4} s^2}{2r^3 s} \right)^{-3} \left( \frac{6r^2 s^3}{8r s^5} \right)^2

Learn how to simplify complex algebraic expressions involving exponents with a detailed breakdown of the solution. Apply exponent rules, simplify terms, and multiply expressions effectively. Suitable for high school students.

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