Let's break down the expression step by step:

Given expression: $\frac{3}{7} \times \left(6 + 8 \times 3 - 2\right) + \left[\frac{1}{5} \div \frac{7}{25} - \left(\frac{3}{7} + \frac{8}{14}\right)\right]$

Step 1: Simplify inside the parentheses and brackets

1. Simplify inside the first parentheses: $6 + 8 \times 3 - 2$

   • Multiplication first: $8 \times 3 = 24$
   • Addition and subtraction: $6 + 24 - 2 = 28$

2. Now, multiply by $\frac{3}{7}$: $\frac{3}{7} \times 28 = \frac{3 \times 28}{7} = \frac{84}{7} = 12$

3. Simplify inside the brackets: $\frac{1}{5} \div \frac{7}{25} - \left(\frac{3}{7} + \frac{8}{14}\right)$

   • First, let's handle the division: $\frac{1}{5} \div \frac{7}{25} = \frac{1}{5} \times \frac{25}{7} = \frac{25}{35} = \frac{5}{7}$

   • Then, simplify the addition inside the parentheses: $\frac{8}{14} = \frac{4}{7} \quad \text{(simplifying the fraction)}$ $\frac{3}{7} + \frac{4}{7} = \frac{7}{7} = 1$

4. Subtract the results inside the brackets: $\frac{5}{7} - 1 = \frac{5}{7} - \frac{7}{7} = \frac{-2}{7}$

Step 2: Add the two main parts together

The final step is to add the results from Step 1: $12 + \left(\frac{-2}{7}\right) = 12 - \frac{2}{7}$

Final Answer

The final expression evaluates to: $\frac{82}{7}$

This is the simplified and final result.

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Do you want more details on any of the steps? Here are some related questions:

1. What are the rules for the order of operations in mathematics?
2. How do you simplify complex fractions?
3. How do you handle operations with negative numbers in algebra?
4. Can you explain how to convert a mixed number to an improper fraction?
5. What are common mistakes to avoid in multi-step arithmetic problems?

Tip: Always work through parentheses and brackets first and follow the order of operations (PEMDAS/BODMAS) to avoid mistakes.