https://file.aiquickdraw.com/mathbot/file/1726541605402xej5lsvx.jpg

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!

[24 - 2 * (5 + 6)^2 + 7 * (10 - 32 / 16)] / [7^2 * (4 + 2^5)]

Learn how to simplify the expression [24 - 2 * (5 + 6)^2 + 7 * (10 - 32 / 16)] / [7^2 * (4 + 2^5)] step-by-step. This problem covers key concepts like order of operations, exponents, and fractions. Suitable for students in Grades 9-10.

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