1. **Simplify the expression:**
2. **Given expression:**
$\frac{1}{2+\sqrt{6}} + \frac{1}{\sqrt{5}+\sqrt{6}} + \frac{1}{\sqrt{6}+\sqrt{7}} + \frac{1}{\sqrt{7}+\sqrt{8}}$
3. **Rationalizing the denominators:**
To simplify the given expression, we need to rationalize the denominators.
- For $\frac{1}{2+\sqrt{6}}$, multiply the numerator and denominator by the conjugate of the denominator: $2-\sqrt{6}$.
This gives us: $\frac{1}{2+\sqrt{6}} \times \frac{2-\sqrt{6}}{2-\sqrt{6}} = \frac{2-\sqrt{6}}{2^2 - (\sqrt{6})^2} = \frac{2-\sqrt{6}}{4-6} = \frac{2-\sqrt{6}}{-2} = \frac{\sqrt{6}-2}{2}$.
- Similarly, rationalize the denominators for the other fractions as well.
4. **Simplified expression:**
After rationalizing the denominators, the expression becomes:
$\frac{\sqrt{6}-2}{2} + \frac{\sqrt{6}-\sqrt{5}}{1} + \frac{\sqrt{7}-\sqrt{6}}{1} + \frac{\sqrt{8}-\sqrt{7}}{1}$
Simplifying further, we get:
$\frac{\sqrt{6}+\sqrt{6}+\sqrt{7}+\sqrt{8}-2-\sqrt{5}-\sqrt{6}-\sqrt{7}}{2}$
Combining like terms, we get:
$\boxed{\frac{\sqrt{8}-\sqrt{5}}{2}}$
5. **Conclusion:**
Therefore, the simplified form of the given expression is $\frac{\sqrt{8}-\sqrt{5}}{2}$.