Let's solve the given exercises from the image. The problems involve simplifying expressions with radicals:

1. Efectueu les següents sumes i restes d'arrels:

Let's solve them step by step.

(a) $6\sqrt{3} - 4\sqrt{3} + 5\sqrt{3}$

• Combine like terms: $(6 - 4 + 5)\sqrt{3} = 7\sqrt{3}$.

(b) $3\sqrt{2} + 5\sqrt{2} - 8\sqrt{2}$

• Combine like terms: $(3 + 5 - 8)\sqrt{2} = 0$.

(c) $3\sqrt{2} - 3\sqrt{8} + 3\sqrt{18}$

• Simplify each term:
  • $3\sqrt{2}$ remains as is.
  • $3\sqrt{8} = 3 \times 2\sqrt{2} = 6\sqrt{2}$.
  • $3\sqrt{18} = 3 \times 3\sqrt{2} = 9\sqrt{2}$.
• Combine the terms: $3\sqrt{2} - 6\sqrt{2} + 9\sqrt{2} = 6\sqrt{2}$.

(d) $2a\sqrt{3} - \sqrt{27a^2} + a\sqrt{12}$

• Simplify the radicals:
  • $\sqrt{27a^2} = 3a$.
  • $a\sqrt{12} = a \times 2\sqrt{3} = 2a\sqrt{3}$.
• Combine the terms: $2a\sqrt{3} + 2a\sqrt{3} - 3a = 4a\sqrt{3} - 3a$.

(e) $2a\sqrt{2} - \sqrt{8} + 3\sqrt{2}$

• Simplify $\sqrt{8} = 2\sqrt{2}$.
• Combine the terms: $2a\sqrt{2} + 3\sqrt{2} - 2\sqrt{2} = 2a\sqrt{2} + \sqrt{2}$.

(f) $(3 + a)\sqrt{5} - \sqrt{125} + \sqrt{5a^2}$

• Simplify the radicals:
  • $\sqrt{125} = 5\sqrt{5}$.
  • $\sqrt{5a^2} = a\sqrt{5}$.
• Combine the terms: $(3 + a)\sqrt{5} - 5\sqrt{5} + a\sqrt{5} = (2 + 2a)\sqrt{5}$.

(g) $\frac{7}{\sqrt{54}} - 3\sqrt{18} + \sqrt{24} - \frac{3}{5}\sqrt{50} - \sqrt{6}$

• Simplify each term:
  • $\sqrt{54} = 3\sqrt{6}$, so $\frac{7}{\sqrt{54}} = \frac{7}{3\sqrt{6}}$.
  • $3\sqrt{18} = 9\sqrt{2}$.
  • $\sqrt{24} = 2\sqrt{6}$.
  • $\frac{3}{5}\sqrt{50} = \frac{3}{5} \times 5\sqrt{2} = 3\sqrt{2}$.
• Combine the terms step by step to get the simplified expression.

(h) $\sqrt{16} - \sqrt{54}$

• Simplify:
  • $\sqrt{16} = 4$.
  • $\sqrt{54} = 3\sqrt{6}$.
• Result: $4 - 3\sqrt{6}$.

Would you like a detailed explanation for each step or any specific problem? Here are related questions:

1. How do you combine radicals with different coefficients?
2. What are the general rules for simplifying square roots?
3. How can you rationalize the denominator in expressions like $\frac{7}{3\sqrt{6}}$?
4. Why is it important to simplify radicals before combining terms?
5. How do you handle radicals with variables?

Tip: When adding or subtracting radicals, they must be "like radicals," meaning the expression under the radical must be identical.