8. \( 4 \sqrt{6}-(-\sqrt{2}-\sqrt{3})^{3}-(-2 \sqrt{3}-\sqrt{2})^{2}+8-18 \sqrt{\frac{1}{3}}-\frac{6}{\sqrt{2}+1}-\sqrt{27} \)

To simplify the expression \[ 8 + 4 \sqrt{6} - (-\sqrt{2} - \sqrt{3})^3 - (-2 \sqrt{3} - \sqrt{2})^2 - 18 \sqrt{\frac{1}{3}} - \frac{6}{\sqrt{2}+1} - \sqrt{27}, \] we will evaluate each term step by step. 1. **Calculate \((- \sqrt{2} - \sqrt{3})^3\)**: \[ (-\sqrt{2} - \sqrt{3})^3 = -(\sqrt{2} + \sqrt{3})^3. \] Using the binomial expansion: \[ (\sqrt{2} + \sqrt{3})^3 = \sqrt{2}^3 + 3\sqrt{2}^2\sqrt{3} + 3\sqrt{2}\sqrt{3}^2 + \sqrt{3}^3 = 2\sqrt{2} + 3 \cdot 2 \cdot \sqrt{3} + 3 \cdot \sqrt{2} \cdot 3 + 3 = 2\sqrt{2} + 6\sqrt{3} + 9. \] Therefore, \[ (-\sqrt{2} - \sqrt{3})^3 = -(2\sqrt{2} + 6\sqrt{3} + 9) = -2\sqrt{2} - 6\sqrt{3} - 9. \] 2. **Calculate \((-2\sqrt{3} - \sqrt{2})^2\)**: \[ (-2\sqrt{3} - \sqrt{2})^2 = (2\sqrt{3} + \sqrt{2})^2 = (2\sqrt{3})^2 + 2(2\sqrt{3})(\sqrt{2}) + (\sqrt{2})^2 = 4 \cdot 3 + 4\sqrt{6} + 2 = 12 + 4\sqrt{6} + 2 = 14 + 4\sqrt{6}. \] 3. **Calculate \(-18 \sqrt{\frac{1}{3}}\)**: \[ -18 \sqrt{\frac{1}{3}} = -18 \cdot \frac{1}{\sqrt{3}} = -6\sqrt{3}. \] 4. **Calculate \(-\frac{6}{\sqrt{2}+1}\)**: To rationalize the denominator: \[ -\frac{6}{\sqrt{2}+1} \cdot \frac{\sqrt{2}-1}{\sqrt{2}-1} = -\frac{6(\sqrt{2}-1)}{(\sqrt{2})^2 - 1^2} = -\frac{6(\sqrt{2}-1)}{2-1} = -6(\sqrt{2}-1) = -6\sqrt{2} + 6. \] 5. **Calculate \(-\sqrt{27}\)**: \[ -\sqrt{27} = -3\sqrt{3}. \] Now, we can substitute these results back into the original expression: \[ 8 + 4\sqrt{6} - (-2\sqrt{2} - 6\sqrt{3} - 9) - (14 + 4\sqrt{6}) - 6\sqrt{3} - 6\sqrt{2} + 6 - 3\sqrt{3}. \] This simplifies to: \[ 8 + 4\sqrt{6} + 2\sqrt{2} + 6\sqrt{3} + 9 - 14 - 4\sqrt{6} - 6\sqrt{3} - 6\sqrt{2} + 6 - 3\sqrt{3}. \] Combining like terms: - Constant terms: \[ 8 + 9 + 6 - 14 = 9. \] - Terms with \(\sqrt{6}\): \[ 4\sqrt{6} - 4\sqrt{6} = 0. \] - Terms with \(\sqrt{2}\): \[ 2\sqrt{2} - 6\sqrt{2} = -4\sqrt{2}. \] - Terms with \(\sqrt{3}\): \[ 6\sqrt{3} - 6\sqrt{3} - 3\sqrt{3} = -3\sqrt{3}. \] Putting it all together, we have: \[ 9 - 4\sqrt{2} - 3\sqrt{3}. \] Thus, the final simplified expression is: \[ \boxed{9 - 4\sqrt{2} - 3\sqrt{3}}. \]