Let's simplify each of the given expressions:

1. $\frac{10}{\sqrt{2}}$

   Rationalize the denominator: $\frac{10}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}$

2. $\frac{\sqrt{5}}{\sqrt{8}}$

   Simplify the expression: $\frac{\sqrt{5}}{\sqrt{8}} = \frac{\sqrt{5}}{2\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{10}}{4}$

3. $\frac{\sqrt{2}}{6\sqrt{3}}$

   Rationalize the denominator: $\frac{\sqrt{2}}{6\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{6}}{18}$

4. $\frac{3}{2\sqrt{6}}$

   Rationalize the denominator: $\frac{3}{2\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{3\sqrt{6}}{12} = \frac{\sqrt{6}}{4}$

5. $\frac{2\sqrt{5}}{3 + \sqrt{5}}$

   Multiply numerator and denominator by the conjugate of the denominator: $\frac{2\sqrt{5}}{3 + \sqrt{5}} \times \frac{3 - \sqrt{5}}{3 - \sqrt{5}} = \frac{2\sqrt{5}(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})} = \frac{6\sqrt{5} - 10}{9 - 5} = \frac{6\sqrt{5} - 10}{4} = \frac{3\sqrt{5} - 5}{2}$

6. $\frac{2\sqrt{10}}{\sqrt{7} - \sqrt{5}}$

   Multiply numerator and denominator by the conjugate of the denominator: $\frac{2\sqrt{10}}{\sqrt{7} - \sqrt{5}} \times \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} + \sqrt{5}} = \frac{2\sqrt{10}(\sqrt{7} + \sqrt{5})}{(\sqrt{7})^2 - (\sqrt{5})^2} = \frac{2\sqrt{10}(\sqrt{7} + \sqrt{5})}{7 - 5} = \sqrt{10}(\sqrt{7} + \sqrt{5})$

7. $\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}$

   Multiply numerator and denominator by the conjugate of the denominator: $\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \times \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{(\sqrt{3} - \sqrt{2})^2}{(\sqrt{3})^2 - (\sqrt{2})^2} = \frac{3 - 2\sqrt{6} + 2}{3 - 2} = 5 - 2\sqrt{6}$

8. $\frac{2\sqrt{2} + \sqrt{7}}{2\sqrt{2} - \sqrt{7}}$

   Multiply numerator and denominator by the conjugate of the denominator: $\frac{2\sqrt{2} + \sqrt{7}}{2\sqrt{2} - \sqrt{7}} \times \frac{2\sqrt{2} + \sqrt{7}}{2\sqrt{2} + \sqrt{7}} = \frac{(2\sqrt{2} + \sqrt{7})^2}{(2\sqrt{2})^2 - (\sqrt{7})^2} = \frac{8 + 4\sqrt{14} + 7}{8 - 7} = 15 + 4\sqrt{14}$

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Do you want further details or have any questions?

Here are 5 questions that expand on the above concepts:

1. How do you multiply and simplify expressions with radicals?
2. What is the purpose of rationalizing the denominator?
3. How does multiplying by the conjugate simplify expressions?
4. Can you show the derivation of a similar radical expression?
5. What are some common mistakes to avoid when working with radicals?

Tip: When dealing with expressions involving square roots, always check if the radicals can be simplified before proceeding with further operations.