a) \(\dfrac{1}{3+\sqrt{2}}+\dfrac{1}{3-\sqrt{2}}\)
\(=\dfrac{3-\sqrt{2}}{\left(3+\sqrt{2}\right)\left(3-\sqrt{2}\right)}+\dfrac{3+\sqrt{2}}{\left(3+\sqrt{2}\right)\left(3-\sqrt{2}\right)}\)
\(=\dfrac{3-\sqrt{2}}{9-2}+\dfrac{3+\sqrt{2}}{9-2}\)
\(=\dfrac{3-\sqrt{2}+3+\sqrt{2}}{8}\)
\(=\dfrac{6}{8}\)
\(=\dfrac{3}{4}\)
b) \(\dfrac{2}{3\sqrt{2}-4}-\dfrac{2}{3\sqrt{2}+4}\)
\(=\dfrac{2\left(3\sqrt{2}+4\right)}{\left(3\sqrt{2}-4\right)\left(3\sqrt{2}+4\right)}-\dfrac{2\left(3\sqrt{2}-4\right)}{\left(3\sqrt{2}+4\right)\left(3\sqrt{2}-4\right)}\)
\(=\dfrac{2\left(3\sqrt{2}+4\right)}{18-16}-\dfrac{2\left(3\sqrt{2}-4\right)}{18-16}\)
\(=\dfrac{2\left(3\sqrt{2}+4\right)}{2}-\dfrac{2\left(3\sqrt{2}-4\right)}{2}\)
\(=3\sqrt{2}+4-3\sqrt{2}+4\)
\(=8\)
c) \(\dfrac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}+\dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}\)
\(=\dfrac{\left(\sqrt{5}-\sqrt{3}\right)^2}{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}+\dfrac{\left(\sqrt{5}+\sqrt{3}\right)^2}{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}\)
\(=\dfrac{5-2\sqrt{15}+3}{5-3}+\dfrac{5+2\sqrt{15}+3}{5-3}\)
\(=\dfrac{8-2\sqrt{15}}{2}+\dfrac{8+2\sqrt{15}}{2}\)
\(=\dfrac{8-2\sqrt{15}+8+2\sqrt{15}}{2}\)
\(=\dfrac{16}{2}\)
\(=8\)
d) \(\dfrac{3}{2\sqrt{2}-3\sqrt{3}}-\dfrac{3}{2\sqrt{2}+3\sqrt{3}}\)
\(=\dfrac{3\left(2\sqrt{2}+3\sqrt{3}\right)}{\left(2\sqrt{2}+3\sqrt{3}\right)\left(2\sqrt{2}-3\sqrt{3}\right)}-\dfrac{3\left(2\sqrt{2}-3\sqrt{3}\right)}{\left(2\sqrt{2}-3\sqrt{3}\right)\left(2\sqrt{2}+3\sqrt{3}\right)}\)
\(=\dfrac{3\left(2\sqrt{2}+3\sqrt{3}\right)}{8-27}-\dfrac{3\left(2\sqrt{2}-3\sqrt{3}\right)}{8-27}\)
\(=\dfrac{6\sqrt{2}+9\sqrt{3}-6\sqrt{2}+9\sqrt{3}}{-19}\)
\(=\dfrac{18\sqrt{3}}{-19}\)
\(=-\dfrac{18\sqrt{3}}{19}\)
