1. **Problem Statement:** Find the exact value of $\cot S$ in simplest form for the right triangle $QRS$ with right angle at $R$. Given sides: $QR=5$, $RS=\sqrt{17}$, and hypotenuse $QS=\sqrt{42}$. 2. **Recall the definition:** $\cot S = \frac{\text{adjacent side to } S}{\text{opposite side to } S}$. 3. **Identify sides relative to angle $S$:** - Opposite side to $S$ is $QR = 5$. - Adjacent side to $S$ is $RS = \sqrt{17}$. 4. **Calculate $\cot S$:** $$\cot S = \frac{RS}{QR} = \frac{\sqrt{17}}{5}$$ 5. **Simplify if possible:** The fraction $\frac{\sqrt{17}}{5}$ is already in simplest form. **Final answer:** $$\boxed{\cot S = \frac{\sqrt{17}}{5}}$$