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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.

Rationalize the denominator of the following radical expressions.

This guide provides solutions to rationalizing the denominator of radical expressions such as \(\frac{10}{\sqrt{2}}\), \(\frac{2\sqrt{5}}{3\sqrt{2}}\), and others. Learn how to apply the technique of multiplying by the conjugate and simplifying expressions effectively. Ideal for Grade 9-10 students.

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