We are asked to simplify the expression $\sqrt{320} - \sqrt{80} + \sqrt{\frac{25}{9}}$.

1. Problem Description

We are asked to simplify the expression $\sqrt{320} - \sqrt{80} + \sqrt{\frac{25}{9}}$.

2. Solution Steps

First, we simplify the radicals $\sqrt{320}$ and $\sqrt{80}$.

$\sqrt{320} = \sqrt{64 \cdot 5} = \sqrt{64} \cdot \sqrt{5} = 8\sqrt{5}$

$\sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4\sqrt{5}$

Then we simplify $\sqrt{\frac{25}{9}}$.

$\sqrt{\frac{25}{9}} = \frac{\sqrt{25}}{\sqrt{9}} = \frac{5}{3}$

Now, we substitute these values into the original expression:

$\sqrt{320} - \sqrt{80} + \sqrt{\frac{25}{9}} = 8\sqrt{5} - 4\sqrt{5} + \frac{5}{3} = (8-4)\sqrt{5} + \frac{5}{3} = 4\sqrt{5} + \frac{5}{3}$

3. Final Answer

$4\sqrt{5} + \frac{5}{3}$