1. **State the problem:** We need to find the value of $\cos(OPA)$ given by the formula $$\cos(OPA) = \frac{x^2 - 8x + 40}{\sqrt{(x^2 - 16x + 80)(x^2 + 100)}}$$ when $x = 8$. 2. **Substitute $x = 8$ into the numerator:** $$8^2 - 8 \times 8 + 40 = 64 - 64 + 40 = 40$$ 3. **Substitute $x = 8$ into the first term inside the square root:** $$8^2 - 16 \times 8 + 80 = 64 - 128 + 80 = 16$$ 4. **Substitute $x = 8$ into the second term inside the square root:** $$8^2 + 100 = 64 + 100 = 164$$ 5. **Calculate the denominator:** $$\sqrt{16 \times 164} = \sqrt{2624}$$ 6. **Simplify $\sqrt{2624}$:** $$2624 = 16 \times 164$$ $$\sqrt{2624} = \sqrt{16} \times \sqrt{164} = 4 \times \sqrt{164}$$ 7. **Write the expression for $\cos(OPA)$ with substituted values:** $$\cos(OPA) = \frac{40}{4 \times \sqrt{164}}$$ 8. **Simplify the fraction by canceling 4:** $$\cos(OPA) = \frac{\cancel{40}^{{10}}}{\cancel{4} \times \sqrt{164}} = \frac{10}{\sqrt{164}}$$ 9. **Rationalize the denominator:** $$\cos(OPA) = \frac{10}{\sqrt{164}} \times \frac{\sqrt{164}}{\sqrt{164}} = \frac{10 \sqrt{164}}{164}$$ 10. **Simplify the fraction:** $$\frac{10}{164} = \frac{5}{82}$$ 11. **Final answer:** $$\cos(OPA) = \frac{5 \sqrt{164}}{82}$$