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.