1. Simplify $8\sqrt{3} - 4\sqrt{3}$. Since both terms have $\sqrt{3}$, subtract the coefficients: $$8 - 4 = 4$$ So, the result is: $$4\sqrt{3}$$ 2. Simplify $8\sqrt{2} - 7\sqrt{2}$. Subtract coefficients: $$8 - 7 = 1$$ Result: $$\sqrt{2}$$ 3. Simplify $4\sqrt{a} + \sqrt{a}$. Add coefficients: $$4 + 1 = 5$$ Result: $$5\sqrt{a}$$ 4. Simplify $-\sqrt{7} + 2\sqrt{7}$. Add coefficients: $$-1 + 2 = 1$$ Result: $$\sqrt{7}$$ 5. Simplify $5^5\sqrt{5} - 10^5\sqrt{5} + 15^5\sqrt{5}$. Here, $5^5 = 3125$, $10^5 = 100000$, $15^5 = 759375$. Combine coefficients: $$3125 - 100000 + 759375 = 662500$$ Result: $$662500\sqrt{5}$$ 6. Simplify $2\sqrt{x} - \sqrt{x} + 3\sqrt{x}$. Add coefficients: $$2 - 1 + 3 = 4$$ Result: $$4\sqrt{x}$$ 7. Simplify $6\sqrt{3} + 7\sqrt{3} - 2\sqrt{3}$. Add coefficients: $$6 + 7 - 2 = 11$$ Result: $$11\sqrt{3}$$ 8. Simplify $\frac{9}{6} - 4\sqrt{6} + \frac{2}{6}$. First, simplify fractions: $$\frac{9}{6} = \frac{3}{2}, \quad \frac{2}{6} = \frac{1}{3}$$ Add fractions: $$\frac{3}{2} + \frac{1}{3} = \frac{9}{6} + \frac{2}{6} = \frac{11}{6}$$ Result: $$\frac{11}{6} - 4\sqrt{6}$$ 9. Simplify $29\sqrt{y} - 13a\sqrt{y} + a\sqrt{y}$. Combine like terms with $a\sqrt{y}$: $$-13a + a = -12a$$ Result: $$29\sqrt{y} - 12a\sqrt{y}$$ 10. Simplify $\frac{1}{2} \sqrt{2} + \frac{1}{3} \sqrt{2}$. Add coefficients: $$\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$$ Result: $$\frac{5}{6} \sqrt{2}$$ 11. Simplify $\frac{1}{2} 4\sqrt{3} - \frac{1}{3} 4\sqrt{3}$. Multiply coefficients: $$\frac{1}{2} \times 4 = 2, \quad \frac{1}{3} \times 4 = \frac{4}{3}$$ Subtract: $$2 - \frac{4}{3} = \frac{6}{3} - \frac{4}{3} = \frac{2}{3}$$ Result: $$\frac{2}{3} \sqrt{3}$$ 12. Simplify $7\sqrt{a} - 3 \times 7\sqrt{a} + 5 \times 7\sqrt{a}$. Calculate coefficients: $$7 - 21 + 35 = 21$$ Result: $$21\sqrt{a}$$ 13. Simplify $3b\sqrt{7} - b\sqrt{7} + c\sqrt{7}$. Combine $b\sqrt{7}$ terms: $$3b - b = 2b$$ Result: $$2b\sqrt{7} + c\sqrt{7}$$ 14. Simplify $5\sqrt{3x} + c\sqrt{3x}$. Add coefficients: $$5 + c$$ Result: $$(5 + c)\sqrt{3x}$$ 15. Simplify $\sqrt{8} - 2\sqrt{8} + \sqrt{8} - \sqrt{8}$. Combine coefficients: $$1 - 2 + 1 - 1 = -1$$ Result: $$-\sqrt{8}$$ Simplify $\sqrt{8} = 2\sqrt{2}$: Final result: $$-2\sqrt{2}$$ 16. Simplify $2 3\sqrt{a} + 7 3\sqrt{a} - 5 3\sqrt{a}$. Calculate coefficients: $$2 + 7 - 5 = 4$$ Result: $$4 3\sqrt{a}$$