a) \( \displaystyle{2 \over {\sqrt 3 - 1}} - {2 \over {\sqrt 3 + 1}}\) \( \displaystyle= {{2(\sqrt 3 + 1) - 2(\sqrt 3 - 1)} \over {(\sqrt 3 + 1)(\sqrt 3 - 1)}}\)
\( \displaystyle = {{2\sqrt 3 + 2 - 2\sqrt 3 + 2} \over {3 - 1}} = {4 \over 2} = 2\)
b) \( \displaystyle{5 \over {12(2\sqrt 5 + 3\sqrt 2 )}} - {5 \over {12(2\sqrt 5 - 3\sqrt 2 )}}\)
\( \displaystyle = {{5(2\sqrt 5 - 3\sqrt 2 ) - 5(2\sqrt 5 + 3\sqrt 2 )} \over {12(2\sqrt 5 + 3\sqrt 2 )(2\sqrt 5 - 3\sqrt 2 )}}\)
\( \displaystyle\eqalign{
& = {{10\sqrt 5 - 15\sqrt 2 - 10\sqrt 5 - 15\sqrt 2 } \over {12(20 - 18)}} \cr
& = {{ - 30\sqrt 2 } \over {12.2}} = - {{5\sqrt 2 } \over 4} \cr} \)
c) \( \displaystyle{{5 + \sqrt 5 } \over {5 - \sqrt 5 }} + {{5 - \sqrt 5 } \over {5 + \sqrt 5 }}\) \( \displaystyle\displaystyle= {{{{(5 + \sqrt 5 )}^2} + {{(5 - \sqrt 5 )}^2}} \over {(5 + \sqrt 5 )(5 - \sqrt 5 )}}\)
\( \displaystyle = {{25 + 10\sqrt 5 + 5 + 25 - 10\sqrt 5 + 5} \over {25 - 5}}\) \( \displaystyle= {{60} \over {20}} = 3\)
d) \( \displaystyle{{\sqrt 3 } \over {\sqrt {\sqrt 3 + 1} - 1}} - {{\sqrt 3 } \over {\sqrt {\sqrt 3 + 1} + 1}}\)
\( \displaystyle = {{\sqrt 3 (\sqrt {\sqrt 3 + 1} + 1) - \sqrt 3 (\sqrt {\sqrt 3 + 1} - 1)} \over {(\sqrt {\sqrt 3 + 1} + 1)(\sqrt {\sqrt 3 + 1} - 1)}}\)
\( \displaystyle\eqalign{
& = {{\sqrt {3(\sqrt 3 + 1)} + \sqrt 3 - \sqrt {3(\sqrt 3 + 1)} + \sqrt 3 } \over {\sqrt 3 + 1 - 1}} \cr
& = {{2\sqrt 3 } \over {\sqrt 3 }} = 2 \cr} \)
