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Let's go through the problems one by one:

### Problem 6:
\[
\frac{3\sqrt{5}}{\sqrt{6} - \sqrt{8}}
\]

To simplify this, multiply both the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{6} + \sqrt{8}\). This eliminates the radicals in the denominator:

\[
\frac{3\sqrt{5}}{\sqrt{6} - \sqrt{8}} \cdot \frac{\sqrt{6} + \sqrt{8}}{\sqrt{6} + \sqrt{8}} = \frac{3\sqrt{5}(\sqrt{6} + \sqrt{8})}{(\sqrt{6})^2 - (\sqrt{8})^2}
\]

The denominator simplifies as follows:
\[
(\sqrt{6})^2 - (\sqrt{8})^2 = 6 - 8 = -2
\]

Now, distribute in the numerator:
\[
3\sqrt{5}(\sqrt{6} + \sqrt{8}) = 3\sqrt{30} + 3\sqrt{40} = 3\sqrt{30} + 6\sqrt{10}
\]

So the expression becomes:
\[
\frac{3\sqrt{30} + 6\sqrt{10}}{-2} = -\frac{3\sqrt{30}}{2} - \frac{6\sqrt{10}}{2} = -\frac{3\sqrt{30}}{2} - 3\sqrt{10}
\]

### Problem 7:
\[
\frac{7}{2\sqrt{3}} + \sqrt{2}
\]

To rationalize the first term, multiply the numerator and denominator by \(\sqrt{3}\):

\[
\frac{7}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{7\sqrt{3}}{2 \cdot 3} = \frac{7\sqrt{3}}{6}
\]

Thus, the expression becomes:
\[
\frac{7\sqrt{3}}{6} + \sqrt{2}
\]

### Problem 8:
\[
\frac{5\sqrt{10}}{\sqrt{7}}
\]

Rationalize the denominator by multiplying both the numerator and denominator by \(\sqrt{7}\):

\[
\frac{5\sqrt{10}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = \frac{5\sqrt{70}}{7}
\]

Thus, the simplified form is:
\[
\frac{5\sqrt{70}}{7}
\]

---

Would you like further clarification on any of these steps?

Here are 5 related questions:
1. How do you multiply conjugates involving square roots?
2. How do you rationalize a denominator with multiple terms?
3. What happens when you multiply square roots with different radicands?
4. How can you simplify radicals like \(\sqrt{40}\) to lowest terms?
5. Why is rationalizing the denominator important in simplifying expressions?

**Tip**: Always check if the radicand (the number inside the square root) can be simplified before rationalizing!

6. \frac{3\sqrt{5}}{\sqrt{6} - \sqrt{8}} 
7. \frac{7}{2\sqrt{3}} + \sqrt{2} 
8. \frac{5\sqrt{10}}{\sqrt{7}}

Learn how to simplify radical expressions like \frac{3\sqrt{5}}{\sqrt{6} - \sqrt{8}}, rationalize square roots, and apply conjugates. Step-by-step solutions for expressions involving radicals, suitable for high school students in Grades 9-12.

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