Let's solve the expression step by step:

Given expression:  
\[
(5-2)^3 \times 2 + \left[-4 + \frac{-7}{(-2+4)^2}\right]
\]

### Step 1: Simplify the parentheses and exponents
1. \( 5 - 2 = 3 \)
2. \( 3^3 = 27 \)
3. \( -2 + 4 = 2 \)
4. \( 2^2 = 4 \)

### Step 2: Substitute back and simplify further
Now substitute these results back into the expression:
\[
27 \times 2 + \left[-4 + \frac{-7}{4}\right]
\]

### Step 3: Perform the multiplication
\[
54 + \left[-4 + \frac{-7}{4}\right]
\]

### Step 4: Simplify the fraction
\[
\frac{-7}{4} = -\frac{7}{4} = -1.75
\]

### Step 5: Add the terms inside the brackets
\[
-4 - 1.75 = -5.75
\]

### Step 6: Final addition
\[
54 + (-5.75) = 54 - 5.75 = 48.25
\]

### Final Answer
\[
\boxed{48.25}
\]

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you simplify expressions with multiple operations?
2. What are the rules for order of operations in math?
3. How can you handle negative numbers in equations?
4. What are some common mistakes when dealing with exponents?
5. How does the distributive property apply in more complex expressions?

**Tip:** Always follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Learn how to solve the expression (5-2)^3 × 2 + [-4 + (-7) ÷ (-2+4)^2] step-by-step with detailed explanations. This problem covers arithmetic operations, exponents, and fractions. Suitable for students in Grades 6-8.

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