Question

Please answer this question fast

Answer 1

Answer:

4

Explanation

It is based on rationalization

Formula used

${x}^{2} - {y}^{2} = (x + y)(x - y) \\ (x + y) {}^{2} = {x}^{2} + {y}^{2} + 2xy$

$\frac{1}{1 + \sqrt{3} + \sqrt{5} } \times \frac{1 - ( \sqrt{3} + \sqrt{5} ) }{1 - ( \sqrt{3} + \sqrt{5}) } \\ \\ = \frac{1 - ( \sqrt{3 }{+ \sqrt{5}) } }{ {1}^{2} - ( \sqrt{3} + \sqrt{5} ) {}^{2} } \\ \\ = \frac{1 - ( \sqrt{3} + \sqrt{5}) }{1 - 8 - 2 \sqrt{15} } \\ \\ = \frac{ \sqrt{3} + \sqrt{5} - 1 }{2 \sqrt{15} + 7 } \times \frac{2 \sqrt{15} - 7 }{2 \sqrt{15} - 7 } \\ \\ = \frac{ (\sqrt{3} + \sqrt{5} - 1)(2 \sqrt{5} \sqrt{3} - 7) }{ {(2 \sqrt{15}) }^{2} - {7}^{2} } \\ \\ = \frac{6 \sqrt{5} + 10 \sqrt{3} - 2 \sqrt{15} - 7 \sqrt{3} - 7 \sqrt{5} + 7 }{60 - 49} \\ \\ = \frac{7 - \sqrt{5} + ..... }{11}$

So Compare , P is non rational term which is 7/11