$\displaystyle \begin{array}{{>{\displaystyle}l}} a) \ \frac{\sqrt{6} +\sqrt{14}}{2\sqrt{3} +\sqrt{28}} =\frac{\sqrt{2} .\sqrt{3} +\sqrt{2} .\sqrt{7}}{2.\sqrt{3} +\sqrt{4} .\sqrt{7}}\\ =\frac{\sqrt{2}\left(\sqrt{3} +\sqrt{7}\right)}{2.\sqrt{3} +2\sqrt{7}} =\frac{\sqrt{2}\left(\sqrt{3} +\sqrt{7}\right)}{2\left(\sqrt{3} +\sqrt{7}\right)}\\ =\frac{\sqrt{2}}{2} \ \\ b) \ \frac{9\sqrt{5} +3\sqrt{27}}{\sqrt{5} +\sqrt{3}} =\frac{9\sqrt{5} +3.\sqrt{9}\sqrt{3}}{\sqrt{5} +\sqrt{3}}\\ =\frac{9\sqrt{5} +9\sqrt{3}}{\sqrt{5} +\sqrt{3}} =\frac{9\left(\sqrt{5} +\sqrt{3}\right)}{\sqrt{5} +\sqrt{3}} =9\\ c) \ \frac{\sqrt{2} +\sqrt{3} +\sqrt{6} +\sqrt{8} +4}{\sqrt{2} +\sqrt{3} +\sqrt{4}}\\ =\frac{\sqrt{2} +1+\sqrt{3}\left( 1+\sqrt{2}\right) +2\sqrt{2} +3}{\sqrt{2} +\sqrt{3} +\sqrt{4}}\\ =\frac{\sqrt{2} +1+\sqrt{3}\left( 1+\sqrt{2}\right) +\left( 1+\sqrt{2}\right)^{2}}{\sqrt{2} +\sqrt{3} +\sqrt{4}}\\ =\frac{\left(\sqrt{2} +1\right)\left( 1+\sqrt{3} +1+\sqrt{2}\right)}{\sqrt{2} +\sqrt{3} +2}\\ =\frac{\left(\sqrt{2} +1\right)\left( 2+\sqrt{3} +\sqrt{2}\right)}{\sqrt{2} +\sqrt{3} +2}\\ =\sqrt{2} +1 \end{array}$
