The given expression is:

$\frac{{(4^2 - 6 \times 2)^2}}{{\left( \frac{4}{2} \times 4 \right)}} + 7$

Let's break it down step-by-step:

Step 1: Simplify the expression inside the parentheses

• $4^2 = 16$
• $6 \times 2 = 12$

So, the expression inside the first parentheses becomes:

$16 - 12 = 4$

Step 2: Square the result of the parentheses

$(4)^2 = 16$

Step 3: Simplify the denominator

• $\frac{4}{2} = 2$
• $2 \times 4 = 8$

So, the denominator becomes 8.

Step 4: Substitute into the main expression

Now, the expression looks like this:

$\frac{16}{8} + 7$

Step 5: Simplify the division

$\frac{16}{8} = 2$

Step 6: Add 7

$2 + 7 = 9$

So, the final answer is 9.

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Would you like a more detailed explanation or have any questions about the steps? Here are 5 related questions you might find interesting:

1. How do parentheses affect the order of operations in mathematics?
2. What is the importance of using the correct order of operations when solving complex expressions?
3. How would the result change if we swapped the positions of the numbers inside the parentheses?
4. What are some common mistakes when handling fractions and exponents together in expressions?
5. How can we use the distributive property in expressions like these for simplification?

Tip: Always start simplifying from inside the parentheses and carefully follow the order of operations (PEMDAS) to avoid mistakes.