1. **State the problem:** Simplify the expression $2a\sqrt{3} - \sqrt{27a^2} + a\sqrt{12}$. 2. **Recall the rules:** - $\sqrt{xy} = \sqrt{x} \times \sqrt{y}$ - $\sqrt{a^2} = |a|$, but here we treat $a$ as positive for simplification. 3. **Simplify each term:** - $2a\sqrt{3}$ stays as is. - $\sqrt{27a^2} = \sqrt{27} \times \sqrt{a^2} = \sqrt{9 \times 3} \times a = 3\sqrt{3}a$. - $a\sqrt{12} = a \times \sqrt{4 \times 3} = a \times 2\sqrt{3} = 2a\sqrt{3}$. 4. **Rewrite the expression:** $$2a\sqrt{3} - 3a\sqrt{3} + 2a\sqrt{3}$$ 5. **Combine like terms:** $$ (2a - 3a + 2a)\sqrt{3} = (2 - 3 + 2)a\sqrt{3} = 1a\sqrt{3} = a\sqrt{3}$$ --- 6. **Next expression:** Simplify $\sqrt{\frac{50}{9}} + \sqrt{\frac{18}{16}} - 5\sqrt{\frac{200}{36}}$. 7. **Simplify each square root:** - $\sqrt{\frac{50}{9}} = \frac{\sqrt{50}}{3} = \frac{\sqrt{25 \times 2}}{3} = \frac{5\sqrt{2}}{3}$. - $\sqrt{\frac{18}{16}} = \frac{\sqrt{18}}{4} = \frac{\sqrt{9 \times 2}}{4} = \frac{3\sqrt{2}}{4}$. - $5\sqrt{\frac{200}{36}} = 5 \times \frac{\sqrt{200}}{6} = 5 \times \frac{\sqrt{100 \times 2}}{6} = 5 \times \frac{10\sqrt{2}}{6} = 5 \times \frac{5\sqrt{2}}{3} = \frac{25\sqrt{2}}{3}$. 8. **Rewrite the expression:** $$\frac{5\sqrt{2}}{3} + \frac{3\sqrt{2}}{4} - \frac{25\sqrt{2}}{3}$$ 9. **Combine like terms by finding common denominator 12:** - $\frac{5\sqrt{2}}{3} = \frac{20\sqrt{2}}{12}$ - $\frac{3\sqrt{2}}{4} = \frac{9\sqrt{2}}{12}$ - $\frac{25\sqrt{2}}{3} = \frac{100\sqrt{2}}{12}$ 10. **Sum:** $$\frac{20\sqrt{2}}{12} + \frac{9\sqrt{2}}{12} - \frac{100\sqrt{2}}{12} = \frac{(20 + 9 - 100)\sqrt{2}}{12} = \frac{-71\sqrt{2}}{12}$$ **Final answers:** - For the first expression: $a\sqrt{3}$ - For the second expression: $-\frac{71\sqrt{2}}{12}$