a)
\( \displaystyle\eqalign{
& {1 \over {\sqrt 3 + \sqrt 2 + 1}}= {1 \over {\sqrt 3 + (\sqrt 2 + 1)}} \cr
& = {{\sqrt 3 - (\sqrt 2 + 1)} \over {\left[ {\sqrt 3 + (\sqrt 2 + 1)} \right]\left[ {\sqrt 3 - (\sqrt 2 + 1)} \right]}} \cr} \)
\( \displaystyle = {{\sqrt 3 - \sqrt 2 - 1} \over {3 - {{(\sqrt 2 + 1)}^2}}} = {{\sqrt 3 - \sqrt 2 - 1} \over {3 - (2 + 2\sqrt 2 + 1)}}\) \( \displaystyle = {{\sqrt 3 - \sqrt 2 - 1} \over { - 2\sqrt 2 }}\)
\( \displaystyle = {{ - \sqrt 2 (\sqrt 3 - \sqrt 2 - 1)} \over {2{{(\sqrt 2 )}^2}}}\) \( \displaystyle = {{ - \sqrt 6 + 2 + \sqrt 2 } \over 4}\)
b)
\( \displaystyle{1 \over {\sqrt 5 - \sqrt 3 + 2}}\) \( \displaystyle = {{\sqrt 5 + (\sqrt 3 - 2)} \over {\left[ {\sqrt 5 - (\sqrt 3 - 2)} \right]\left[ {\sqrt 5 + (\sqrt 3 - 2)} \right]}}\)
\( \displaystyle = {{\sqrt 5 + (\sqrt 3 - 2)} \over {5 - {{(\sqrt 3 - 2)}^2}}}\) \( \displaystyle = {{\sqrt 5 + (\sqrt 3 - 2)} \over {5 - (3 - 4\sqrt 3 + 4)}}\) \( \displaystyle = {{\sqrt 5 + (\sqrt 3 - 2)} \over {4\sqrt 3 - 2}}\)
\( \displaystyle= {{\sqrt 5 + \sqrt 3 - 2} \over {2(2\sqrt 3 - 1)}}\) \( \displaystyle = {{(\sqrt 5 + \sqrt 3 - 2)(2\sqrt 3 + 1)} \over {2\left[ {(2\sqrt 3 - 1)(2\sqrt 3 + 1)} \right]}}\)
\( \displaystyle = {{2\sqrt {15} + \sqrt 5 + 6 + \sqrt 3 - 4\sqrt 3 - 2} \over {2(12 - 1)}} \)
\( \displaystyle= {{2\sqrt {15} + \sqrt 5 + 4 - 3\sqrt 3 } \over {22}} \)
