1. **State the problem:** Express $\tan K$ as a fraction in simplest terms given the values related to triangle sides or angles. 2. **Recall the definition:** $\tan K = \frac{\text{opposite side}}{\text{adjacent side}}$. 3. **Identify the sides:** From the problem, assume opposite side to angle $K$ is $\sqrt{26}$ and adjacent side is $2\sqrt{30}$. 4. **Write the fraction:** $$\tan K = \frac{\sqrt{26}}{2\sqrt{30}}$$ 5. **Simplify the fraction:** $$\tan K = \frac{\sqrt{26}}{2\sqrt{30}} = \frac{\sqrt{26}}{2\sqrt{30}} \times \frac{\sqrt{30}}{\sqrt{30}} = \frac{\sqrt{26} \times \sqrt{30}}{2 \times 30} = \frac{\sqrt{780}}{60}$$ 6. **Simplify $\sqrt{780}$:** $$\sqrt{780} = \sqrt{4 \times 195} = 2\sqrt{195}$$ 7. **Substitute back:** $$\tan K = \frac{2\sqrt{195}}{60} = \frac{\cancel{2}\sqrt{195}}{\cancel{60}30} = \frac{\sqrt{195}}{30}$$ 8. **Final answer:** $$\boxed{\tan K = \frac{\sqrt{195}}{30}}$$