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If x= 2(sq.rt of 2 + sq.rt of 6)/3{sq.rt of(2+sq.rt of 3)} then simplify x to get x as a rational number.

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\(=\frac{2(\sqrt 2+\sqrt 6)}{3\sqrt{2+\sqrt 3}}\)

\(=\frac{2(\sqrt 2+\sqrt 6)}{3\sqrt{2+\sqrt 3}}\times \frac{\sqrt {2-\sqrt 3}}{\sqrt{2-\sqrt 3}}\)

\(=\frac{2(\sqrt 2+\sqrt 6)\times \sqrt{2-\sqrt 3}}{3\sqrt{4-3}}\)   \((\because (a+b)(a-b)=a^2-b^2)\)

\(=\frac{2}{3}\left(\sqrt{(2-\sqrt 3)\times (\sqrt 2+\sqrt 6)^2}\right)\)

\(=\frac{2}{3}\sqrt{(2-\sqrt 3)(2+6+4\sqrt 3)}\)

\(=\frac{2}{3}\sqrt{16-8\sqrt 3+8\sqrt 3-12}\)

\(=\frac{2}{3}\sqrt 4\) \(=\frac{2}{3}\times 2\) \(=\frac{4}{3}\).

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