The problem asks for the result of the following expression:

$\left( \frac{2^3}{3^2} \right) \times \left( \frac{3^5}{2^4} \right)^2$

Let's break this down step by step:

Step 1: Simplify the powers inside the parentheses.

• The first part is $\frac{2^3}{3^2}$, which stays as it is for now.
• The second part is $\left( \frac{3^5}{2^4} \right)^2$. Applying the exponent to both the numerator and the denominator gives us:

$\frac{3^{5 \times 2}}{2^{4 \times 2}} = \frac{3^{10}}{2^8}$

Step 2: Multiply both terms.

Now multiply:

$\left( \frac{2^3}{3^2} \right) \times \left( \frac{3^{10}}{2^8} \right)$

We can combine the powers of 2 and 3:

$\frac{2^3 \times 3^{10}}{3^2 \times 2^8} = \frac{2^{3-8} \times 3^{10-2}}{1} = \frac{2^{-5} \times 3^8}{1}$

This simplifies to:

$\frac{3^8}{2^5}$

Step 3: Calculate the powers.

• $3^8 = 6561$
• $2^5 = 32$

So, the expression becomes:

$\frac{6561}{32} \approx 204.4$

However, the answers provided are whole numbers, so this might indicate a mistake in the process of simplification or interpretation.

Let me know if you'd like more clarification on any step! Here are 5 related questions:

1. How do we apply exponents to fractions correctly?
2. What are the rules for multiplying terms with different bases?
3. Can you explain how to combine exponents when multiplying fractions?
4. What is the difference between negative exponents and positive exponents?
5. How do powers of two and three affect the results in these expressions?

Tip: Always break down complex expressions step by step, applying exponent rules carefully to avoid mistakes.