Let's break down the expression step by step:

We are asked to evaluate:

$6 \div \left[ 5 \times (4 - 2) \div (5^2 + 3) \right] \div 2$

Step 1: Simplify inside the parentheses

1. $4 - 2 = 2$
2. $5^2 = 25$, so $25 + 3 = 28$

Now the expression becomes:

$6 \div \left[ 5 \times 2 \div 28 \right] \div 2$

Step 2: Simplify the multiplication and division inside the brackets

• $5 \times 2 = 10$
• $10 \div 28 \approx 0.357$

Now the expression becomes:

$6 \div 0.357 \div 2$

Step 3: Perform the division

First, divide $6 \div 0.357$:

$6 \div 0.357 \approx 16.798$

Now divide by 2:

$16.798 \div 2 \approx 8.399$

Final Answer:

$\boxed{8.399}$

Would you like further details or explanations on any part of the calculation?

Related Questions:

1. How would the result change if you used different values for the numbers inside the parentheses?
2. What would the result be if the operation inside the brackets was subtraction instead of division?
3. How do parentheses affect the order of operations in more complex expressions?
4. What is the importance of rounding in mathematical calculations?
5. How do you deal with long decimals when performing division?

Tip: When dealing with parentheses, always simplify the expressions inside them first, then proceed with the remaining operations following the correct order (PEMDAS).