x \(=\frac{2(\sqrt 2+\sqrt 6)}{3\sqrt{2+\sqrt 3}}\)
\(=\frac{2(\sqrt 2+\sqrt 6)}{3\sqrt{2+\sqrt 3}}\times \frac{\sqrt {2-\sqrt 3}}{\sqrt{2-\sqrt 3}}\)
\(=\frac{2(\sqrt 2+\sqrt 6)\times \sqrt{2-\sqrt 3}}{3\sqrt{4-3}}\) \((\because (a+b)(a-b)=a^2-b^2)\)
\(=\frac{2}{3}\left(\sqrt{(2-\sqrt 3)\times (\sqrt 2+\sqrt 6)^2}\right)\)
\(=\frac{2}{3}\sqrt{(2-\sqrt 3)(2+6+4\sqrt 3)}\)
\(=\frac{2}{3}\sqrt{16-8\sqrt 3+8\sqrt 3-12}\)
\(=\frac{2}{3}\sqrt 4\) \(=\frac{2}{3}\times 2\) \(=\frac{4}{3}\).
