1. **Stating the problem:** Simplify the expression $$\frac{2}{3} \sqrt{18} + 2 \sqrt{27} - \sqrt{108} + 0.3 \sqrt{200}$$. 2. **Recall the rule:** Simplify square roots by factoring out perfect squares: $$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$$. 3. **Simplify each term:** - $$\sqrt{18} = \sqrt{9 \times 2} = 3 \sqrt{2}$$ - $$\sqrt{27} = \sqrt{9 \times 3} = 3 \sqrt{3}$$ - $$\sqrt{108} = \sqrt{36 \times 3} = 6 \sqrt{3}$$ - $$\sqrt{200} = \sqrt{100 \times 2} = 10 \sqrt{2}$$ 4. **Substitute back:** $$\frac{2}{3} \times 3 \sqrt{2} + 2 \times 3 \sqrt{3} - 6 \sqrt{3} + 0.3 \times 10 \sqrt{2}$$ 5. **Simplify coefficients:** - $$\frac{2}{3} \times 3 = 2$$ - $$2 \times 3 = 6$$ - $$0.3 \times 10 = 3$$ Expression becomes: $$2 \sqrt{2} + 6 \sqrt{3} - 6 \sqrt{3} + 3 \sqrt{2}$$ 6. **Combine like terms:** - For $$\sqrt{2}$$ terms: $$2 \sqrt{2} + 3 \sqrt{2} = 5 \sqrt{2}$$ - For $$\sqrt{3}$$ terms: $$6 \sqrt{3} - 6 \sqrt{3} = 0$$ 7. **Final answer:** $$5 \sqrt{2}$$