https://file.aiquickdraw.com/mathbot/file/17273626230180m32jnvp.jpg

The image contains problems related to square roots and simplifications. I will work through the following two sections in detail:

### 13. Να κάνετε τις πράξεις:  
This requires simplifying the following expressions.

**(α)**  
\[
\sqrt{2} (\sqrt{8} + \sqrt{18})
\]  
- Simplify each square root:  
\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}, \quad \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}
\]  
- Substitute back into the expression:  
\[
\sqrt{2}(2\sqrt{2} + 3\sqrt{2}) = \sqrt{2} \times 5\sqrt{2}
\]  
- Apply multiplication:  
\[
= 5 \times \sqrt{2 \times 2} = 5 \times 2 = 10
\]

**(β)**  
\[
\sqrt{8} - 3 \sqrt{200} - \sqrt{50}
\]  
- Simplify each square root:  
\[
\sqrt{8} = 2\sqrt{2}, \quad \sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2}, \quad \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}
\]  
- Substitute back into the expression:  
\[
2\sqrt{2} - 3 \times 10\sqrt{2} - 5\sqrt{2} = 2\sqrt{2} - 30\sqrt{2} - 5\sqrt{2}
\]  
- Combine like terms:  
\[
(2 - 30 - 5)\sqrt{2} = -33\sqrt{2}
\]

---

### 14. Να αποδείξετε ότι:  
These are proof exercises.

**(α)**  
\[
\sqrt{2} \cdot \sqrt{20} = 2\sqrt{10}
\]  
- Simplify the left-hand side:  
\[
\sqrt{2 \times 20} = \sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}
\]  
Thus, the left-hand side equals the right-hand side.

**(β)**  
\[
\frac{\sqrt{3} \cdot \sqrt{6}}{\sqrt{75}} = 3\sqrt{2}
\]  
- Simplify each term:  
\[
\sqrt{3} \cdot \sqrt{6} = \sqrt{18} = 3\sqrt{2}, \quad \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}
\]  
- Substitute into the expression:  
\[
\frac{3\sqrt{2}}{5\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{6}}{15} = \frac{\sqrt{6}}{5}
\]  

Would you like to explore these or other parts of the image in more detail? Here are a few related questions for practice:

1. Can you simplify \(\sqrt{12} + \sqrt{27}\)?
2. What is the value of \(\sqrt{18} - \sqrt{8}\)?
3. Show that \(\sqrt{48} = 4\sqrt{3}\).
4. Simplify \(\frac{\sqrt{50}}{\sqrt{2}}\).
5. Prove that \(\sqrt{5} \cdot \sqrt{20} = 10\).

**Tip:** When simplifying square roots, look for perfect squares as factors, which makes the process faster.

Math problems involving square root simplifications, including operations and proofs, from the uploaded image.

This page covers the simplification of square roots and proofs of radical identities, with detailed examples and explanations. Problems include simplifying expressions like √2(√8 + √18) and proving √2 * √20 = 2√10, suitable for students in Grades 9-11.

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