Trục căn thức ở mẫu:

a) \(\frac{{4 + 3\sqrt 5 }}{{\sqrt 5 }};\)

b) \(\frac{1}{{\sqrt 5 - 2}};\)

c) \(\frac{{3 + \sqrt 3 }}{{1 - \sqrt 3 }};\)

d) \(\frac{{\sqrt 2 }}{{\sqrt 3 + \sqrt 2 }}.\)

a) \(\frac{{4 + 3\sqrt 5 }}{{\sqrt 5 }} = \frac{{\left( {4 + 3\sqrt 5 } \right)\sqrt 5 }}{{{{\left( {\sqrt 5 } \right)}^2}}} = \frac{{4\sqrt 5 + 15}}{5}.\)

b) \(\frac{1}{{\sqrt 5 - 2}} = \frac{{\sqrt 5 + 2}}{{\left( {\sqrt 5 - 2} \right)\left( {\sqrt 5 + 2} \right)}}\)

\( = \frac{{\sqrt 5 + 2}}{{{{\left( {\sqrt 5 } \right)}^2} - {2^2}}} = \frac{{\sqrt 5 + 2}}{{5 - 4}} = \sqrt 5 + 2.\)

c) \(\frac{{3 + \sqrt 3 }}{{1 - \sqrt 3 }} = \frac{{3{{\left( {1 + \sqrt 3 } \right)}^2}}}{{\left( {1 - \sqrt 3 } \right)\left( {1 + \sqrt 3 } \right)}} = \frac{{3\left( {1 + 2\sqrt 3 + 3} \right)}}{{{1^2} - {{\left( {\sqrt 3 } \right)}^2}}}\)

\( = \frac{{3 + 6\sqrt 3 + 9}}{{1 - 3}} = \frac{{12 + 6\sqrt 3 }}{{ - 2}} = - 6 - 3\sqrt 3 .\)

d) \(\frac{{\sqrt 2 }}{{\sqrt 3 + \sqrt 2 }} = \frac{{\sqrt 2 \left( {\sqrt 3 - \sqrt 2 } \right)}}{{\left( {\sqrt 3 + \sqrt 2 } \right)\left( {\sqrt 3 - \sqrt 2 } \right)}}\)

\( = \frac{{\sqrt 6 - \sqrt 4 }}{{{{\left( {\sqrt 3 } \right)}^2} - {{\left( {\sqrt 2 } \right)}^2}}} = \frac{{\sqrt 6 - 2}}{{3 - 2}} = \sqrt 6 - 2.\)