Question

The answers for question 16

Answer 1

16). I)
$\frac{1}{ \sqrt{3} + \sqrt{2} - 1 } \times \frac{ \sqrt{3} + \sqrt{2} + 1}{ \sqrt{3} + \sqrt{2 } + 1} \\ = \frac{ \sqrt{3} + \sqrt{2} + 1 }{ {( \sqrt{3} + \sqrt{2} )}^{2} - {1}^{2} } \\ = \frac{ \sqrt{3} + \sqrt{2} + 1 }{3 + 2 + 2 \sqrt{6} - 1} \\ = \frac{ \sqrt{3} + \sqrt{2} + 1 }{4 + 2 \sqrt{6} } \\ = \frac{( \sqrt{3} + \sqrt{2} + 1)(4 - 2 \sqrt{6)} }{16 - 24} \\ = \frac{4 \sqrt{3} - 2 \sqrt{18} + 4 \sqrt{2} - 2 \sqrt{12} + 4 - 2 \sqrt{6} }{ - 8} \\ = \frac{4 \sqrt{3} - 6 \sqrt{2} + 4 \sqrt{2} - 4 \sqrt{3} + 4 - 2 \sqrt{6} }{ - 8} \\ = \frac{4 - 2 \sqrt{2} - 2 \sqrt{6} }{ - 8} \\ = \frac{2 - \sqrt{2} - \sqrt{6} }{ - 4} \\ = \frac{ \sqrt{2} + \sqrt{6} - 2 }{4}$


$= \frac{1}{3 - 2 \sqrt{2} + \sqrt{5} } \times \frac{3 - 2 \sqrt{2} - \sqrt{5} }{3 - 2 \sqrt{2} - \sqrt{5} } \\ = \frac{3 - 2 \sqrt{2} - \sqrt{5} }{( {3 - 2 \sqrt{2}) }^{2} - ( { \sqrt{5}) }^{2} } \\ = \frac{3 - 2 \sqrt{2} - \sqrt{5} }{9 + 8 - 12 \sqrt{2} - 5 } \\ = \frac{3 - 2 \sqrt{2} - \sqrt{5} }{12 - 12 \sqrt{2} } \\ = \frac{3 - 2 \sqrt{2} - \sqrt{5} }{12(1 - \sqrt{2} )} \times \frac{1 + \sqrt{2} }{1 + \sqrt{2} } \\ = \frac{3 + 3 \sqrt{2} - 2 \sqrt{2} - 4 - \sqrt{5} - \sqrt{10}}{12(1 - 2)} \\ = \frac{ - 1 + \sqrt{2} - \sqrt{5} - \sqrt{10} }{ - 12} \\ = \frac{1 + \sqrt{5} + \sqrt{10} - \sqrt{2} }{12}$