$\displaystyle \begin{array}{{>{\displaystyle}l}} a.\ \frac{1}{3+\sqrt{2}} +\frac{1}{3-\sqrt{2}} =\frac{3-\sqrt{2} +3+\sqrt{2}}{\left( 3+\sqrt{2}\right)\left( 3-\sqrt{2}\right)} =\frac{6}{9-2} =\frac{6}{7}\\ b.\ \frac{2}{3\sqrt{2} -4} +\frac{2}{3\sqrt{2} +4} =\frac{2\left( 3\sqrt{2} +4+3\sqrt{2} -4\right)}{\left( 3\sqrt{2} -4\right)\left( 3\sqrt{2} +4\right)} =\frac{12\sqrt{2}}{9.2-16} =6\sqrt{2}\\ c.\ \frac{\sqrt{5} -\sqrt{3}}{\sqrt{5} +\sqrt{3}} +\frac{\sqrt{5} +\sqrt{3}}{\sqrt{5} -\sqrt{3}} =\frac{\left(\sqrt{5} -\sqrt{3}\right)^{2} +\left(\sqrt{5} +\sqrt{3}\right)^{2}}{\left(\sqrt{5} -\sqrt{3}\right)\left(\sqrt{5} -\sqrt{3}\right)} =\frac{8-\sqrt{15} +8+\sqrt{15}}{5-3} =12\\ d.\ \frac{3}{2\sqrt{2} -3\sqrt{3}} -\frac{3}{2\sqrt{2} +3\sqrt{3}} =\frac{3\left( 2\sqrt{2} +3\sqrt{3} -2\sqrt{2} +3\sqrt{3}\right)}{8-27} =-\frac{18}{19}\sqrt{3} \end{array}$
