1. Stating the problem: Calculate the value of the expression $25 - (-16 \times 8 \div 9)$.\n\n2. Recall the order of operations: multiplication and division are performed before addition and subtraction, and operations are done from left to right.\n\n3. First, calculate the multiplication and division inside the parentheses: $-16 \times 8 \div 9$.\n\n4. Calculate $-16 \times 8 = -128$.\n\n5. Now divide $-128 \div 9$. We write this as $\frac{-128}{9}$.\n\n6. The expression becomes $25 - \left(-\frac{128}{9}\right)$.\n\n7. Subtracting a negative is the same as adding a positive, so $25 - \left(-\frac{128}{9}\right) = 25 + \frac{128}{9}$.\n\n8. Convert 25 to a fraction with denominator 9: $25 = \frac{225}{9}$.\n\n9. Add the fractions: $\frac{225}{9} + \frac{128}{9} = \frac{225 + 128}{9} = \frac{353}{9}$.\n\n10. The final answer is $\frac{353}{9}$, which is approximately $39.22$.