The problem is to simplify the expression $\sqrt{20} + 5\sqrt{2} - \sqrt{\frac{5}{9}} + \sqrt{50}$.

1. Problem Description

The problem is to simplify the expression $\sqrt{20} + 5\sqrt{2} - \sqrt{\frac{5}{9}} + \sqrt{50}$.

2. Solution Steps

First, simplify each term:

$\sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}$

$\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$

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

Now, substitute the simplified terms back into the original expression:

$2\sqrt{5} + 5\sqrt{2} - \frac{\sqrt{5}}{3} + 5\sqrt{2}$

Combine like terms:

$(2\sqrt{5} - \frac{\sqrt{5}}{3}) + (5\sqrt{2} + 5\sqrt{2})$

$(\frac{6\sqrt{5}}{3} - \frac{\sqrt{5}}{3}) + (10\sqrt{2})$

$\frac{5\sqrt{5}}{3} + 10\sqrt{2}$

3. Final Answer

$\frac{5\sqrt{5}}{3} + 10\sqrt{2}$