The value of
$$\left(\sqrt[6]{27} - \sqrt{6\frac{3}{4}}\right)^2 + \frac{2(\sqrt{2} + \sqrt{6})}{3\sqrt{2} + \sqrt{3}}$$ is

Expression : $$\left(\sqrt[6]{27} - \sqrt{6\frac{3}{4}}\right)^2 + \frac{2(\sqrt{2} + \sqrt{6})}{3\sqrt{2} + \sqrt{3}}$$

= $$[(3)^{\frac{3}{6}}-(\frac{3\sqrt3}{2})]^2$$ $$+\frac{2(\sqrt{2} + \sqrt{6})}{3\sqrt{2} + \sqrt{3}}$$

= $$(\sqrt3-\frac{3\sqrt3}{2})^2+$$ $$\frac{2(\sqrt{2} + \sqrt{6})}{3\sqrt{2} + \sqrt{3}}$$

= $$(\frac{-\sqrt3}{2})^2+$$ $$\frac{2(\sqrt{2} + \sqrt{6})}{3\sqrt{2} + \sqrt{3}}$$

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

= $$\frac{9\sqrt2+3\sqrt3+8\sqrt2+8\sqrt6}{4(3\sqrt2+\sqrt3)}$$

= $$\frac{3\sqrt3(\sqrt6+1)+8\sqrt2(\sqrt3+1)}{4\sqrt3(\sqrt6+1)}$$