$\displaystyle \begin{array}{{>{\displaystyle}l}} \sqrt{\frac{\sqrt{5}}{8\sqrt{5} +3\sqrt{35}}}\left( 3\sqrt{2} +\sqrt{14}\right)\\ \sqrt{\frac{\sqrt{5}}{\sqrt{5}\left( 8+3\sqrt{7}\right)}} \ .\left[\sqrt{2}\left( 3+\sqrt{7}\right)\right]\\ =\frac{\sqrt{2}\left( 3+\sqrt{7}\right)}{\sqrt{8+3\sqrt{7}}} =\frac{2\sqrt{2}\left( 3+\sqrt{7}\right)}{\sqrt{16+6\sqrt{7}}} \ \\ =\frac{2\sqrt{2}\left( 3+\sqrt{7}\right)}{\sqrt{9+2.3\sqrt{7} +7}} =\frac{2\sqrt{2}\left( 3+\sqrt{7}\right)}{\sqrt{\left( 3+\sqrt{7}\right)^{2}}}\\ =\frac{2\sqrt{2}\left( 3+\sqrt{7}\right)}{3+\sqrt{7}} =2\sqrt{2}\\ \end{array}$