jmc

First, we'll divide $\sqrt{5}$ into each term in the numerator of the fraction inside the big radical: $$\sqrt{\frac{\frac{5}{\sqrt{80}}+\frac{\sqrt{845}}{9}+\sqrt{45}}{\sqrt{5}}}=\sqrt{\frac{5}{\sqrt{80}\cdot \sqrt{5}}+\frac{\sqrt{845}}{9\sqrt{5}}+\frac{\sqrt{45}}{\sqrt{5}}}.$$Let's attack each fraction within the square root separately. First, $$\frac{5}{\sqrt{80}\cdot \sqrt{5}}=\frac{5}{\sqrt{400}}=\frac{5}{20}=\frac{1}{4}.$$The second one is trickier: $$\frac{\sqrt{845}}{9\sqrt{5}}=\frac{\sqrt{169}}{9}=\frac{13}{9}.$$Finally, $\frac{\sqrt{45}}{\sqrt{5}}=\sqrt{9}=3$. Adding these together, we get $$\sqrt{\frac{1}{4}+\frac{13}{9}+3}=\sqrt{\frac{9+52+108}{36}}=\sqrt{\frac{169}{36}}=\boxed{\frac{13}{6}}.$$