Question

$\frac{1}{1 - \sqrt{2 + \sqrt{3} } }$
Rationalise the denominator

Answer 1

$\frac{ 1 - \sqrt{3} + \sqrt{2 + \sqrt{3} } - \sqrt{2 \sqrt{3} + 3} }{2}$

Step-by-step explanation:

$\frac{1}{1 - \sqrt{2 + \sqrt{3} } } \\ = \frac{1}{1 - \sqrt{2 + \sqrt{3} } } \times \frac{1 + \sqrt{2 + \sqrt{3} } }{1 + \sqrt{2 + \sqrt{3} } } \\ = \frac{1 + \sqrt{2 + \sqrt{3} } }{ {1}^{2} - ( \sqrt{2 + \sqrt{3} } )^{2} } \\ = \frac{1 + \sqrt{2 + \sqrt{3} } }{1 - 2 - \sqrt{3} } \\ = \frac{1 + \sqrt{2 + \sqrt{3} } }{ - 1- \sqrt{3} } \\ = \frac{1 + \sqrt{2 + \sqrt{3} } }{ - 1- \sqrt{3} } \times \frac{ - 1 + \sqrt{3} }{ - 1 + \sqrt{3} } \\ = \frac{ - 1 + \sqrt{3} - \sqrt{2 + \sqrt{3} } + \sqrt{3}. \sqrt{2 + \sqrt{3} } }{( - 1)^{2} - ( \sqrt{3} )^{2} } \\ = \frac{ - 1 + \sqrt{3} - \sqrt{2 + \sqrt{3} } + \sqrt{2 \sqrt{3} + 3} }{1 - 3} \\ = \frac{ - 1 + \sqrt{3} - \sqrt{2 + \sqrt{3} } + \sqrt{2 \sqrt{3} + 3} }{ - 2} \\ = \frac{ 1 - \sqrt{3} + \sqrt{2 + \sqrt{3} } - \sqrt{2 \sqrt{3} + 3} }{2}$