https://file.aiquickdraw.com/mathbot/file/1730067348738e7uxhyh3.jpg

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.

Simplify and solve the given expressions with radicals.

Learn how to simplify and solve expressions with radicals in this step-by-step guide. Covers combining like terms and simplifying square roots, with detailed explanations suitable for high school students in Grades 9-11.

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