Determine the exact value of \( \left(4^{2}-2 \times 3+25^{\circ}\right)^{-2} \). Express your answer as a fraction.

To solve the expression \( \left(4^{2}-2 \times 3+25^{\circ}\right)^{-2} \), we first need to break down the components inside the parentheses. 1. Calculate \( 4^2 \): \[ 4^2 = 16 \] 2. Calculate \( 2 \times 3 \): \[ 2 \times 3 = 6 \] 3. Now, substitute these values into the expression inside the parentheses: \[ 4^{2} - 2 \times 3 + 25^{\circ} = 16 - 6 + 25^{\circ} \] 4. Next, simplify this expression: \[ 16 - 6 = 10 \] Since \( 25^{\circ} \) typically represents a degree measure and doesn’t seem to fit into standard arithmetic, we will consider \( 25^{\circ} \) as simply \( 25 \) for the expression. The calculations now continue as: \[ 10 + 25 = 35 \] 5. Now, we can rewrite the expression as: \[ (35)^{-2} \] 6. Calculate \( (35)^{-2} \): \[ (35)^{-2} = \frac{1}{35^2} \] 7. Finally, calculate \( 35^2 \): \[ 35^2 = 1225 \] Thus, the value of the expression is: \[ \frac{1}{1225} \] Therefore, the exact value of \( \left(4^{2}-2 \times 3+25^{\circ}\right)^{-2} \) is \( \boxed{\frac{1}{1225}} \).