Let's break down and solve the given expression step by step.

The expression is:

$\frac{[24 - 2 \cdot (5 + 6)^2] + 7 \cdot (10 - \frac{32}{16})}{7^2 \cdot (4 + 2^5)}$

Step 1: Solve inside the parentheses and exponents.

• $5 + 6 = 11$
• $(5 + 6)^2 = 11^2 = 121$
• $\frac{32}{16} = 2$
• $10 - 2 = 8$
• $2^5 = 32$

Now the expression becomes:

$\frac{[24 - 2 \cdot 121] + 7 \cdot 8}{7^2 \cdot (4 + 32)}$

Step 2: Simplify multiplication and addition.

• $2 \cdot 121 = 242$
• $24 - 242 = -218$
• $7 \cdot 8 = 56$
• $7^2 = 49$
• $4 + 32 = 36$

Now the expression becomes:

$\frac{-218 + 56}{49 \cdot 36}$

Step 3: Simplify the numerator and denominator.

• $-218 + 56 = -162$
• $49 \cdot 36 = 1764$

Now the expression becomes:

$\frac{-162}{1764}$

Step 4: Simplify the fraction.

$\frac{-162}{1764} = \frac{-1}{10.888}$

The approximate result is:

$-0.091$

Thus, the final answer is approximately -0.091.

Would you like more details on any of the steps?

5 Related Questions:

1. How do you simplify fractions like $\frac{-162}{1764}$?
2. What are the rules for exponentiation when applied within parentheses?
3. How do you handle division and multiplication inside a fraction?
4. How can we simplify expressions with powers of two, like $2^5$?
5. What is the role of the order of operations (PEMDAS) in complex expressions like this?

Tip: Always handle parentheses and exponents first when simplifying expressions involving multiple operations!