1. **State the problem:** We need to find the exact value of $\cos Z$ in simplest radical form for the right triangle with sides $YX=\sqrt{31}$ (adjacent to angle $Z$), $YZ=\sqrt{18}$ (opposite to angle $Z$), and hypotenuse $XZ=7$. 2. **Recall the cosine definition:** For an angle in a right triangle, $\cos$ is the ratio of the length of the adjacent side to the hypotenuse: $$\cos Z = \frac{\text{adjacent}}{\text{hypotenuse}}$$ 3. **Identify the sides:** Adjacent side to $Z$ is $YX=\sqrt{31}$, hypotenuse is $XZ=7$. 4. **Write the ratio:** $$\cos Z = \frac{\sqrt{31}}{7}$$ 5. **Check if simplification is possible:** $\sqrt{31}$ is already in simplest radical form and 7 is a prime number, so the fraction is simplest. 6. **Final answer:** $$\boxed{\cos Z = \frac{\sqrt{31}}{7}}$$