a: \(\dfrac{\sqrt{3}+1}{\sqrt{3}-1}+\dfrac{\sqrt{3}-1}{\sqrt{3}+1}\)
\(=\dfrac{\left(\sqrt{3}+1\right)^2+\left(\sqrt{3}-1\right)^2}{\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}\)
\(=\dfrac{4+2\sqrt{3}+4-2\sqrt{3}}{3-1}=\dfrac{8}{2}=4\)
b: \(\dfrac{\sqrt{3+\sqrt{5}}}{\sqrt{2}}-\dfrac{\sqrt{5}-1}{2}\)
\(=\dfrac{\sqrt{6+2\sqrt{5}}}{2}-\dfrac{\sqrt{5}-1}{2}\)
\(=\dfrac{\sqrt{5}+1-\left(\sqrt{5}-1\right)}{2}=\dfrac{2}{2}=1\)
c: \(\dfrac{3-\sqrt{7}}{3+\sqrt{7}}-\dfrac{3+\sqrt{7}}{3-\sqrt{7}}\)
\(=\dfrac{\left(3-\sqrt{7}\right)^2-\left(3+\sqrt{7}\right)^2}{\left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right)}\)
\(=\dfrac{16-6\sqrt{7}-16-6\sqrt{7}}{9-7}=\dfrac{-12\sqrt{7}}{2}=-6\sqrt{7}\)
d: \(\dfrac{\sqrt{3}-\sqrt{6}}{1-\sqrt{2}}-\dfrac{2+\sqrt{8}}{1+\sqrt{2}}\)
\(=\dfrac{\sqrt{3}\left(1-\sqrt{2}\right)}{1-\sqrt{2}}-\dfrac{2\left(1+\sqrt{2}\right)}{1+\sqrt{2}}\)
\(=\sqrt{3}-2\)
e: \(\dfrac{5+7\sqrt{5}}{\sqrt{5}}+\dfrac{11+\sqrt{11}}{1+\sqrt{11}}\)
\(=\dfrac{\sqrt{5}\left(7+\sqrt{5}\right)}{\sqrt{5}}+\dfrac{\sqrt{11}\left(\sqrt{11}+1\right)}{\sqrt{11}}\)
\(=\sqrt{5}+7+\sqrt{11}+1=\sqrt{5}+\sqrt{11}+8\)
f: \(\left(2+\dfrac{3+\sqrt{3}}{\sqrt{3}+1}\right)\left(2-\dfrac{3-\sqrt{3}}{\sqrt{3}-1}\right)\)
\(=\left(2+\dfrac{\sqrt{3}\left(\sqrt{3}+1\right)}{\sqrt{3}+1}\right)\left(2-\dfrac{\sqrt{3}\left(\sqrt{3}-1\right)}{\sqrt{3}-1}\right)\)
\(=\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)=4-3=1\)
