The Simplified value of 6 + 2 2 + 2 + 3 - 6 - 2 2 - 2 - 3 - 2 2 2 + 2

Answer: Option D

$$\begin{array}{rl}& \frac{\sqrt{6}+2}{\sqrt{2}+\sqrt{2+\sqrt{3}}}-\frac{\sqrt{6}-2}{\sqrt{2}-\sqrt{2-\sqrt{3}}}-\frac{2\sqrt{2}}{2+\sqrt{2}}\\ & \Rightarrow \frac{\sqrt{6}+2}{\sqrt{2}+\frac{\sqrt{3}+1}{\sqrt{2}}}-\frac{\sqrt{6}-2}{\sqrt{2}-\frac{\sqrt{3}-1}{\sqrt{2}}}-\frac{2}{\sqrt{2}+1}\\ & \Rightarrow \frac{\left(\sqrt{6}+2\right)\sqrt{2}}{2+\sqrt{3}+1}-\frac{\left(\sqrt{6}-2\right)\sqrt{2}}{2-\sqrt{3}+1}-\frac{2}{\sqrt{2}+1}\\ & \Rightarrow \frac{\sqrt{2}}{\sqrt{3}}\left[\frac{\sqrt{6}+2}{\left(\sqrt{3}+1\right)}-\frac{\sqrt{6}-2}{\left(\sqrt{3}-1\right)}\right]-\frac{2}{\sqrt{2}+1}\times \frac{\sqrt{2}-1}{\sqrt{2}-1}\\ & \Rightarrow \frac{\sqrt{2}}{\sqrt{3}}\left[\frac{\sqrt{18}+2\sqrt{3}-\sqrt{6}-2-\sqrt{18}+2\sqrt{3}-\sqrt{6}+2}{3-1}\right]-\frac{2\left(\sqrt{2}-1\right)}{2-1}\\ & \Rightarrow \frac{\sqrt{2}}{\sqrt{3}}\left[\frac{2\left(-\sqrt{6}+2\sqrt{3}\right)}{2}\right]-2\left(\sqrt{2}-1\right)\\ & \Rightarrow -\sqrt{3}\times \sqrt{2}\times \frac{\sqrt{2}}{\sqrt{3}}+2\sqrt{3}\times \frac{\sqrt{2}}{\sqrt{3}}-2\left(\sqrt{2}-1\right)\\ & \Rightarrow -2+2\sqrt{2}-2\sqrt{2}+2\\ & \Rightarrow 0\end{array}$$