The expression you've provided is:

$\frac{9 \cdot (2^3 - 5) + 2 \cdot (3^2 - 3 \cdot 4)}{7 + 7^2 - (6 - 7)^2}$

I'll go through it step-by-step:

Step 1: Simplify the Numerator

1. $2^3 = 8$, so $2^3 - 5 = 8 - 5 = 3$.
2. $3^2 = 9$ and $3 \cdot 4 = 12$, so $3^2 - 3 \cdot 4 = 9 - 12 = -3$.
3. Substitute these values into the numerator: $9 \cdot 3 + 2 \cdot (-3) = 27 - 6 = 21$

Step 2: Simplify the Denominator

1. $7^2 = 49$, so $7 + 7^2 = 7 + 49 = 56$.
2. $6 - 7 = -1$, so $(6 - 7)^2 = (-1)^2 = 1$.
3. Substitute these values into the denominator: $56 - 1 = 55$

Step 3: Combine the Results

Now the expression simplifies to: $\frac{21}{55}$

Thus, the simplified answer is:

$\frac{21}{55}$

Would you like further explanation or have any questions?

Here are some related questions to explore:

1. What happens if we change one of the numbers in the expression?
2. How can we simplify expressions with different types of operations?
3. Why is the order of operations important in mathematics?
4. How do exponents affect the outcome in complex expressions?
5. What if we apply this expression to a real-world scenario?

Tip: Always follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) to avoid mistakes in calculations.