smc

Squaring the expression and taking the positive square root (since numerator and denominator are positive) of the result yields $$\sqrt{{\left(\frac{2(\sqrt{2}+\sqrt{6})}{3\sqrt{2+\sqrt{3}}}\right)}^2}$$ $$\sqrt{\frac{4(2+4\sqrt{3}+6)}{9(2+\sqrt{3})}}$$ $$\sqrt{\frac{4(8+4\sqrt{3})}{9(2+\sqrt{3})}}$$ $$\sqrt{\frac{16}{9}}$$ $$\frac{4}{3}$$ The answer is $\boxed{\frac{4}{3}}$.