8. - 12/(sqrt(5) + 1)

9. 3/(2sqrt(3) - 3)

10. 6/(sqrt(8) - sqrt(5))

Let's simplify each of these expressions by rationalizing the denominators.

### 8. Simplify: \(-\frac{12}{\sqrt{5} + 1}\)

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:

\[
-\frac{12}{\sqrt{5} + 1} \times \frac{\sqrt{5} - 1}{\sqrt{5} - 1} = -\frac{12(\sqrt{5} - 1)}{(\sqrt{5} + 1)(\sqrt{5} - 1)}
\]

Since \((\sqrt{5} + 1)(\sqrt{5} - 1)\) is a difference of squares:

\[
(\sqrt{5})^2 - (1)^2 = 5 - 1 = 4
\]

Thus:

\[
-\frac{12(\sqrt{5} - 1)}{4} = -3(\sqrt{5} - 1) = -3\sqrt{5} + 3
\]

### 9. Simplify: \(\frac{3}{2\sqrt{3} - 3}\)

Multiply the numerator and denominator by the conjugate of the denominator:

\[
\frac{3}{2\sqrt{3} - 3} \times \frac{2\sqrt{3} + 3}{2\sqrt{3} + 3} = \frac{3(2\sqrt{3} + 3)}{(2\sqrt{3})^2 - (3)^2}
\]

Calculate the denominator:

\[
(2\sqrt{3})^2 - 3^2 = 4 \cdot 3 - 9 = 12 - 9 = 3
\]

Thus:

\[
\frac{3(2\sqrt{3} + 3)}{3} = 2\sqrt{3} + 3
\]

### 10. Simplify: \(\frac{6}{\sqrt{8} - \sqrt{5}}\)

Multiply the numerator and denominator by the conjugate of the denominator:

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

Calculate the denominator:

\[
(\sqrt{8})^2 - (\sqrt{5})^2 = 8 - 5 = 3
\]

Thus:

\[
\frac{6(\sqrt{8} + \sqrt{5})}{3} = 2(\sqrt{8} + \sqrt{5}) = 2\sqrt{8} + 2\sqrt{5} = 4\sqrt{2} + 2\sqrt{5}
\]

### Final Simplified Results:
1. \( -3\sqrt{5} + 3 \)
2. \( 2\sqrt{3} + 3 \)
3. \( 4\sqrt{2} + 2\sqrt{5} \)

Would you like more details or have any questions about these steps?

Here are some related questions:

1. How do you rationalize the denominator in a fraction?
2. What is the difference between conjugates and difference of squares?
3. How can you use conjugates to simplify complex fractions?
4. What are some common pitfalls when rationalizing denominators?
5. How do you apply these techniques to more complex expressions?

**Tip:** Always double-check each step when simplifying, especially when working with conjugates and radicals, to avoid mistakes.

Explore how to simplify complex rational expressions with detailed step-by-step solutions. This page covers rationalizing denominators, conjugates, and difference of squares methods. Ideal for advanced high school or college-level students.

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