6. 1/(sqrt(6) - 2)
7. 7/(sqrt(18) + 3)
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 denominator.

### 6. \( \frac{1}{\sqrt{6} - 2} \)

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator, \( \sqrt{6} + 2 \):

\[
\frac{1}{\sqrt{6} - 2} \times \frac{\sqrt{6} + 2}{\sqrt{6} + 2} = \frac{\sqrt{6} + 2}{(\sqrt{6})^2 - 2^2} = \frac{\sqrt{6} + 2}{6 - 4} = \frac{\sqrt{6} + 2}{2} = \frac{\sqrt{6}}{2} + 1
\]

### 7. \( \frac{7}{\sqrt{18} + 3} \)

Rationalize by multiplying by the conjugate \( \sqrt{18} - 3 \):

\[
\frac{7}{\sqrt{18} + 3} \times \frac{\sqrt{18} - 3}{\sqrt{18} - 3} = \frac{7(\sqrt{18} - 3)}{(\sqrt{18})^2 - 3^2} = \frac{7(\sqrt{18} - 3)}{18 - 9} = \frac{7(\sqrt{18} - 3)}{9} = \frac{7\sqrt{18}}{9} - \frac{21}{9} = \frac{7\sqrt{18}}{9} - \frac{7}{3}
\]

Since \( \sqrt{18} = 3\sqrt{2} \):

\[
\frac{7\sqrt{18}}{9} = \frac{7 \times 3\sqrt{2}}{9} = \frac{7\sqrt{2}}{3}
\]

Final result:

\[
\frac{7\sqrt{2}}{3} - \frac{7}{3} = \frac{7(\sqrt{2} - 1)}{3}
\]

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

Rationalize by multiplying by the conjugate \( \sqrt{5} - 1 \):

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

Explore how to simplify rational expressions involving square roots with detailed step-by-step solutions. This includes rationalizing denominators using conjugates and simplifying expressions. Suitable for high school students.

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