Question

(4 + under root 5 /4 - under root 5 ) +( 4 -root under 5/ 4 + root under5) . sinplify

Answer 1

$\frac{4 + \sqrt{5} }{4 - \sqrt{5} } + \frac{4 - \sqrt{5} }{4 + \sqrt{5} }$

On rationalizing the denominator we get,

$= \frac{4 + \sqrt{5} }{4 - \sqrt{5} } \times \frac{4 + \sqrt{5} }{4 + \sqrt{5} } + \frac{4 - \sqrt{5} }{4 + \sqrt{5} } \times \frac{4 - \sqrt{5} }{4 - \sqrt{5} } \\ \\ = \frac{ {(4)}^{2} + {( \sqrt{5} )}^{2} + 2(4)( \sqrt{5} )}{ {(4)}^{2} - {( \sqrt{5}) }^{2} } + \frac{ {(4)}^{2} + {( \sqrt{5}) }^{2} - 2(4)( \sqrt{5}) }{ {(4)}^{2} - {( \sqrt{5}) }^{2} } \\ \\ = \frac{16 + 5 + 8 \sqrt{5} }{16 - 5} + \frac{16 + 5 - 8 \sqrt{5} }{16 - 5} \\ \\ = \frac{21 + 8 \sqrt{5} }{11} + \frac{21 - 8 \sqrt{5} }{11} \\ \\ = \frac{21 + 8 \sqrt{5} + 21 - 8 \sqrt{5} }{11} \\ \\ = \frac{42}{11}$