[tex] \frac{1}{2 + \sqrt{3} } = \frac{2 - \sqrt{3} }{ {2}^{2} - { \sqrt{3} }^{2} } = \frac{2 - \sqrt{3} }{4 - 3} = \frac{2 - \sqrt{3} }{1} = 2 - \sqrt{3} [/tex]
[tex] \frac{1}{3 + 2 \sqrt{3} } = \frac{1(3 - 2 \sqrt{3}) }{(3 + 2 \sqrt{3}) \times (3 - 2 \sqrt{3}) } = \frac{3 - 2 \sqrt{3} }{9 - 4 \times 3} = \frac{3 - 2 \sqrt{3} }{9 - 12} = \frac{3 - 2 \sqrt{3} }{ - 3} = \frac{ - (2 \sqrt{3} - 3) }{ - 3} = \frac{2 \sqrt{3} - 3 }{3} [/tex]
[tex] \frac{3}{ \sqrt{5} - \sqrt{2} } = \frac{3 \sqrt{5} + 3 \sqrt{2} }{ { \sqrt{5} }^{2} - { \sqrt{2} }^{2} } = \frac{3( \sqrt{5} + \sqrt{2}) }{5 - 2} = \frac{3( \sqrt{5} + \sqrt{2}) }{3} = \sqrt{5} + \sqrt{2} [/tex]
[tex] \frac{4}{2 \sqrt{3} - \sqrt{10} } = \frac{4(2 \sqrt{3} + \sqrt{10}) }{(2 \sqrt{3} - \sqrt{10}) \times (2 \sqrt{3} + 10) } = \frac{4(2 \sqrt{3} + \sqrt{10}) }{4 \times 3 - 10} = \frac{4(2 \sqrt{3} + \sqrt{10}) }{12 - 10} = \frac{4(2 \sqrt{3} + \sqrt{10}) }{2} = 2(2 \sqrt{3} + \sqrt{10}) = 2 \times 2 \sqrt{3} + \sqrt{10} = 4 \sqrt{3} + 2 \sqrt{10} [/tex]
