Question

4+√5/4-√5 + 4-√5/4+√5​

Answer 1

According to question:

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

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

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

$= \frac{ {(4 + \sqrt{5} })^{2} }{ {(4)}^{2} - {( \sqrt{5} })^{2} }$

$= \frac{ {(4)}^{2} + {( \sqrt{5} })^{2} + 2(4)( \sqrt{5}) }{16 - 5}$

$= \frac{16 + 5 + 8 \sqrt{5} }{11}$

$= \frac{21 + 8 \sqrt{5} }{11}$

Now:

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

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

$= \frac{ {(4 - \sqrt{5} })^{2} }{ {(4)}^{2} - {( \sqrt{5} })^{2} }$

$= \frac{ {(4)}^{2} + {( \sqrt{5} })^{2} - 2(4)( \sqrt{5} ) }{16 - 5}$

$= \frac{16 + 5 - 8 \sqrt{5} }{11}$

$= \frac{21 - 8 \sqrt{5} }{11}$

According to question:-

$= \frac{21 + 8 \sqrt{5} }{11} + \frac{21 - 8 \sqrt{5} }{11}$

$= \frac{21 + 8 \sqrt{5} + 21 - 8 \sqrt{5} }{11}$

$= \frac{42}{11}$

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