Solving the Mathematical Expression

This solution demonstrates how to find the value of the given mathematical expression using the standard order of operations. This order helps ensure consistency in calculations and is often remembered by acronyms like PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right) or BODMAS (Brackets, Orders, Division and Multiplication from left to right, Addition and Subtraction from left to right).

The expression we need to evaluate is:

$67 − 9 × [ 4 + 3 × { 16 − 8 ( 6 − 5 ) × 2} ÷ 32]$

Step-by-Step Calculation Breakdown

We will solve this step-by-step, following the order of operations:

1. Parentheses/Brackets (Innermost First): Start with the innermost parentheses: $( 6 − 5 )$.

   $ 6 - 5 = 1 $

   Substitute this back into the expression:

   $67 − 9 × [ 4 + 3 × { 16 − 8 × 1 × 2} ÷ 32]$

2. Multiplication within Curly Braces: Next, perform the multiplication inside the curly braces: $8 × 1 × 2$.

   $ 8 \times 1 \times 2 = 16 $

   The expression now looks like this:

   $67 − 9 × [ 4 + 3 × { 16 − 16} ÷ 32]$

3. Subtraction within Curly Braces: Calculate the subtraction inside the curly braces: ${ 16 − 16 }$.

   $ 16 - 16 = 0 $

   Update the expression:

   $67 − 9 × [ 4 + 3 × 0 ÷ 32]$

4. Multiplication/Division within Brackets (Left to Right): Inside the square brackets, we have addition, multiplication, and division. We perform multiplication and division next. First, $3 × 0$.

   $ 3 \times 0 = 0 $

   The expression becomes:

   $67 − 9 × [ 4 + 0 ÷ 32]$

5. Division within Brackets: Now perform the division: $0 ÷ 32$.

   $ 0 \div 32 = 0 $

   The expression simplifies to:

   $67 − 9 × [ 4 + 0 ]$

6. Addition within Brackets: Perform the addition inside the square brackets: $[ 4 + 0 ]$.

   $ 4 + 0 = 4 $

   The expression is now simpler:

   $67 − 9 × 4$

7. Multiplication: Perform the multiplication outside the brackets: $9 × 4$.

   $ 9 \times 4 = 36 $

   The expression reduces to:

   $67 − 36$

8. Subtraction: Finally, perform the last operation, subtraction.

   $ 67 - 36 = 31 $

Final Value Determination

By carefully applying the order of operations (PEMDAS/BODMAS), we systematically simplify the complex expression. Each step reduces the complexity while maintaining mathematical accuracy.

Therefore, the final value of the expression $67 − 9 × [ 4 + 3 × { 16 − 8 ( 6 − 5 ) × 2} ÷ 32]$ is 31.