1. **State the problem:** Simplify the expression $$\sqrt{\frac{8}{25}} - 3\sqrt{2} - \frac{\sqrt{8}}{2} - \sqrt{882}$$. 2. **Recall important rules:** - $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$ - $$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$$ - Simplify radicals by factoring out perfect squares. 3. **Simplify each term:** - $$\sqrt{\frac{8}{25}} = \frac{\sqrt{8}}{\sqrt{25}} = \frac{\sqrt{4 \cdot 2}}{5} = \frac{2\sqrt{2}}{5}$$ - $$3\sqrt{2}$$ stays as is. - $$\frac{\sqrt{8}}{2} = \frac{\sqrt{4 \cdot 2}}{2} = \frac{2\sqrt{2}}{2} = \sqrt{2}$$ - $$\sqrt{882} = \sqrt{441 \cdot 2} = \sqrt{441} \cdot \sqrt{2} = 21\sqrt{2}$$ 4. **Rewrite the expression with simplified terms:** $$\frac{2\sqrt{2}}{5} - 3\sqrt{2} - \sqrt{2} - 21\sqrt{2}$$ 5. **Combine like terms (all have $$\sqrt{2}$$):** $$\left(\frac{2}{5} - 3 - 1 - 21\right) \sqrt{2}$$ 6. **Calculate the coefficient:** $$\frac{2}{5} - 3 - 1 - 21 = \frac{2}{5} - 25 = \frac{2}{5} - \frac{125}{5} = \frac{2 - 125}{5} = \frac{-123}{5}$$ 7. **Final simplified expression:** $$-\frac{123}{5} \sqrt{2}$$