The image shows a set of problems that ask to rationalize the denominators of various radical expressions. Let's go through them one by one to rationalize the denominators:

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

To rationalize: $\frac{10}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}$

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

Multiply both the numerator and the denominator by $\sqrt{3}$: $\frac{\sqrt{5}}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{15}}{6}$

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

Multiply both the numerator and the denominator by $\sqrt{6}$: $\frac{\sqrt{3}}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{18}}{6} = \frac{3\sqrt{2}}{6} = \frac{\sqrt{2}}{2}$

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

This is equivalent to: $\frac{3}{2\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{3\sqrt{6}}{12} = \frac{\sqrt{6}}{4}$

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

Multiply both the numerator and the denominator by $\sqrt{2}$: $\frac{2\sqrt{5}}{3\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{10}}{6} = \frac{\sqrt{10}}{3}$

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

Multiply both the numerator and the denominator by $\sqrt{6}$: $\frac{\sqrt{3} - \sqrt{2}}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{18} - \sqrt{12}}{6} = \frac{3\sqrt{2} - 2\sqrt{3}}{6}$

7. $\frac{5 - \sqrt{3}}{2\sqrt{5}}$

Multiply both the numerator and the denominator by $\sqrt{5}$: $\frac{5 - \sqrt{3}}{2\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{5\sqrt{5} - \sqrt{15}}{10}$

8. $\frac{2\sqrt{3} - \sqrt{7}}{5\sqrt{2}}$

Multiply both the numerator and the denominator by $\sqrt{2}$: $\frac{2\sqrt{3} - \sqrt{7}}{5\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{6} - \sqrt{14}}{10}$

Would you like a detailed explanation of any specific step? Let me know if you have any questions.

Here are five related questions to extend your understanding:

1. How do you rationalize a denominator with two terms, such as $\frac{a + b}{\sqrt{c} + \sqrt{d}}$?
2. What is the importance of rationalizing denominators in simplifying expressions?
3. Can you rationalize the denominator of $\frac{4}{\sqrt{2} + 1}$?
4. How do you handle the rationalization when the denominator is a cube root?
5. What is the connection between rationalizing denominators and simplifying radical expressions in general?

Tip: When rationalizing denominators with binomials, multiplying by the conjugate of the denominator simplifies the expression effectively.