Let's solve the given expression using the properties of indices.

The expression is:

$\frac{2^3 \cdot \left(3^3 \cdot 2^{-2}\right)^2}{\left(3^{-2} \cdot 2^4\right)^3}$

Step 1: Expand the powers inside the parentheses

Starting with the numerator: $2^3 \cdot \left(3^3 \cdot 2^{-2}\right)^2 = 2^3 \cdot \left(3^{3 \times 2} \cdot 2^{-2 \times 2}\right) = 2^3 \cdot 3^6 \cdot 2^{-4}$

For the denominator: $\left(3^{-2} \cdot 2^4\right)^3 = 3^{-2 \times 3} \cdot 2^{4 \times 3} = 3^{-6} \cdot 2^{12}$

Step 2: Combine like terms in the numerator and denominator

Now, combine the powers of 2 and 3 in the numerator: $2^3 \cdot 2^{-4} \cdot 3^6 = 2^{3 - 4} \cdot 3^6 = 2^{-1} \cdot 3^6 = \frac{3^6}{2}$

And simplify the denominator: $3^{-6} \cdot 2^{12}$

Step 3: Combine the numerator and denominator

Now, we divide the simplified numerator by the simplified denominator: $\frac{\frac{3^6}{2}}{3^{-6} \cdot 2^{12}} = \frac{3^6}{2} \times \frac{1}{3^{-6} \cdot 2^{12}} = \frac{3^6 \times 3^6}{2 \cdot 2^{12}} = \frac{3^{6+6}}{2^{1+12}} = \frac{3^{12}}{2^{13}}$

Final Answer

The expression simplifies to:

$\frac{3^{12}}{2^{13}}$

Would you like any more details or further questions? Here are some related questions you might find interesting:

1. How do you simplify expressions with negative exponents?
2. What is the significance of the base in exponential expressions?
3. How do you expand and simplify expressions with multiple powers?
4. Can you explain the laws of exponents used in solving this problem?
5. How do you handle fractional exponents in expressions?

Tip: Always remember that when you multiply terms with the same base, you add the exponents, and when you divide them, you subtract the exponents.