$\frac{1}{\sqrt{5} + \sqrt{2}} + \frac{1}{\sqrt{8} + \sqrt{5}} + \frac{1}{\sqrt{11} + \sqrt{8}} + ... + \frac{1}{\sqrt{50} + \sqrt{47}} = \frac{\sqrt{5} - \sqrt{2}}{(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})} + \frac{\sqrt{8} - \sqrt{5}}{(\sqrt{8} + \sqrt{5})(\sqrt{8} - \sqrt{5})} + \frac{\sqrt{11} - \sqrt{8}}{(\sqrt{11} + \sqrt{8})(\sqrt{11} - \sqrt{8})} + ... + \frac{\sqrt{50} - \sqrt{47}}{(\sqrt{50} + \sqrt{47})(\sqrt{50} - \sqrt{47})} = \frac{\sqrt{5} - \sqrt{2}}{5 - 2} + \frac{\sqrt{8} - \sqrt{5}}{8 - 5} + \frac{\sqrt{11} - \sqrt{8}}{11 - 8} + ... + \frac{\sqrt{50} - \sqrt{47}}{50 - 47} = \frac{\sqrt{5} - \sqrt{2}}{3} + \frac{\sqrt{8} - \sqrt{5}}{3} + \frac{\sqrt{11} - \sqrt{8}}{3} + ... + \frac{\sqrt{50} - \sqrt{47}}{3} = \frac{\sqrt{5} - \sqrt{2} + \sqrt{8} - \sqrt{5} + \sqrt{11} - \sqrt{8} + ... + \sqrt{50} - \sqrt{47}}{3} = \frac{-\sqrt{2} + \sqrt{50}}{3} = \frac{\sqrt{50} - \sqrt{2}}{3} = \frac{\sqrt{25 * 2} - \sqrt{2}}{3} = \frac{5\sqrt{2} - \sqrt{2}}{3} = \frac{4\sqrt{2}}{3}$