Simplifying the Mathematical Expression

The question asks us to find the simplified value of the mathematical expression: $140 \div \left[\frac{8}{4} \times \left\{10 + 6 - (8 + 5 - (4 + 7))\right\}\right]$ To solve this, we need to follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Step-by-Step Calculation

1. Innermost Parentheses

First, let's calculate the value inside the innermost parentheses: $(4 + 7) = 11$

2. Next Level of Parentheses

Now, substitute this value back into the next set of parentheses: $(8 + 5 - (4 + 7)) = (8 + 5 - 11)$ $(13 - 11) = 2$

3. Braces Calculation

Next, we solve the expression within the braces \{\}: $\{10 + 6 - (8 + 5 - (4 + 7))\} = \{10 + 6 - 2\}$ $\{16 - 2\} = 14$

4. Fraction Calculation

Calculate the value of the fraction $\frac{8}{4}$: $\frac{8}{4} = 2$

5. Multiplication within Brackets

Now, perform the multiplication inside the main brackets []: $\frac{8}{4} \times \left\{10 + 6 - (8 + 5 - (4 + 7))\right\} = 2 \times 14$ $2 \times 14 = 28$

6. Final Division

Finally, perform the division: $140 \div \left[\frac{8}{4} \times \left\{10 + 6 - (8 + 5 - (4 + 7))\right\}\right] = 140 \div 28$ To calculate $140 \div 28$: $140 \div 28 = 5$

Conclusion

Therefore, the simplified value of the given mathematical expression is 5.