https://file.aiquickdraw.com/mathbot/file/1727114541895f6kqoq8y.jpg

Let's solve the given expression step by step.

The expression is:

\[
\frac{12 + 3^2(14 - 2^3) + 6 \cdot 10}{6 + 2^3(3 - 8)^2 - 4^4 + 2^6}
\]

### Step 1: Simplify each part.
#### Numerator:

1. \( 3^2 = 9 \)
2. \( 2^3 = 8 \), so \( 14 - 2^3 = 14 - 8 = 6 \)
3. Multiply: \( 9 \cdot 6 = 54 \)
4. Add: \( 12 + 54 + 6 \cdot 10 = 12 + 54 + 60 = 126 \)

#### Denominator:

1. \( 2^3 = 8 \)
2. \( 3 - 8 = -5 \), and \( (-5)^2 = 25 \)
3. Multiply: \( 8 \cdot 25 = 200 \)
4. Powers: \( 4^4 = 256 \), and \( 2^6 = 64 \)
5. Total denominator: \( 6 + 200 - 256 + 64 = 270 - 256 = 14 \)

### Step 2: Final division.

\[
\frac{126}{14} = 9
\]

Thus, the solution to the expression is \( \boxed{9} \).

Would you like any additional details or explanations?

Here are some related questions to explore:
1. How do powers affect the order of operations in expressions like these?
2. What is the significance of squaring a negative number in expressions?
3. How do parentheses influence the calculation of complex expressions?
4. How can exponents change the magnitude of results?
5. What strategies help in simplifying large expressions efficiently?

**Tip**: Always simplify the exponents and parentheses first when solving complex algebraic expressions.

Solve: (12 + 3^2(14 - 2^3) + 6 * 10) / (6 + 2^3(3 - 8)^2 - 4^4 + 2^6)

Learn how to solve a complex algebraic expression step-by-step using order of operations (PEMDAS) and exponentiation. The problem involves simplifying expressions with exponents, multiplication, and division. Suitable for students in Grades 8-10.

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