sqrt 2 and 3

$\begin{aligned} x & = \sqrt{2+\sqrt 3} \times \sqrt{2+\sqrt{2+\sqrt 3}} \times \sqrt{2+\sqrt{2+\sqrt{2+\sqrt 3}}} \times \sqrt{2-\sqrt{2+\sqrt{2+\sqrt 3}}} \\ & = \sqrt{2+\sqrt 3} \times \sqrt{2+\sqrt{2+\sqrt 3}} \times \sqrt{2^2-\left(\sqrt{2+\sqrt{2+\sqrt 3}}\right)^2} \\ & = \sqrt{2+\sqrt 3} \times \sqrt{2+\sqrt{2+\sqrt 3}} \times \sqrt{2-\sqrt{2+\sqrt 3}} \\ & = \sqrt{2+\sqrt 3} \times \sqrt{2^2-\left(\sqrt{2+\sqrt 3}\right)^2} \\ & = \sqrt{2+\sqrt 3} \times \sqrt{2-\sqrt 3} \\ & = \sqrt{2^2-\left(\sqrt 3\right)^2} \\ & = \sqrt{4-3} \\ & = \boxed{1} \end{aligned}$

Recall that $\begin{aligned}\tan75° =2 +\sqrt 3\end{aligned}$ and $\tan15^{\circ} = 2-\sqrt3$ . Rewriting the given expression as $\begin{aligned} & x = \sqrt{\tan75°} \sqrt{2 + \sqrt{\tan75°}}\sqrt{2 + \sqrt{2+\sqrt{\tan75°}}}\sqrt{2-\sqrt{2+\sqrt{\tan75°}}} \\& x = \sqrt{2\tan{75°} + \tan 75°} \sqrt{2^2 - (\sqrt{2 + \sqrt{\tan75°}}})^2 \\& x =\sqrt{2\tan75° +\tan^2 75°} \sqrt{2 - \sqrt{\tan 75°}} \\& x = \sqrt{\tan75°} \sqrt{\tan{15°}} \\& x = \sqrt{\tan75°}\sqrt{\cot75°} =1 \end{aligned}$