1. The problem is to simplify the expression $$\frac{1}{3}\sqrt{8} + \frac{1}{8} - \frac{1}{5}\sqrt{50}$$. 2. Recall that $$\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$$ and $$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$$. 3. Substitute these into the expression: $$\frac{1}{3} \times 2\sqrt{2} + \frac{1}{8} - \frac{1}{5} \times 5\sqrt{2} = \frac{2}{3}\sqrt{2} + \frac{1}{8} - \sqrt{2}$$. 4. Combine like terms involving $$\sqrt{2}$$: $$\frac{2}{3}\sqrt{2} - \sqrt{2} + \frac{1}{8} = \left(\frac{2}{3} - 1\right)\sqrt{2} + \frac{1}{8}$$. 5. Simplify the coefficient: $$\frac{2}{3} - 1 = \frac{2}{3} - \frac{3}{3} = -\frac{1}{3}$$. 6. So the expression becomes: $$-\frac{1}{3}\sqrt{2} + \frac{1}{8}$$. 7. This is the simplified form of the expression. Final answer: $$-\frac{1}{3}\sqrt{2} + \frac{1}{8}$$.