Question

If a=
$\frac{2 + \sqrt{3} }{2 - \sqrt{3} }$
and
$b = \frac{2 - \sqrt{3} }{2 + \sqrt{3} }$
$then \: the \: value \: ofa + b \: =$
​

Answer 1

$\huge\star\:Solution$

Step-by-step explanation:

$a = \frac{2 + \sqrt{3} }{2 - \sqrt{3} } = \frac{2 + \sqrt{3} }{2 - \sqrt{3} } \times \frac{2 + \sqrt{3} }{2 + \sqrt{3} } \\ a = \frac{({2 + \sqrt{3} })^{2}}{( {2} )^{2} - ({ \sqrt{3} })^{2} } = \frac{ {2}^{2} + { (\sqrt{3}) }^{2} + 4 \sqrt{3} }{4 - 3} \\ a = \frac{4 + 3 + 4 \sqrt{3} }{1} = 7 + 4 \sqrt{3}$

$b = \frac{2 - \sqrt{3} }{2 + \sqrt{3} } = \frac{2 - \sqrt{3} }{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ b = \frac {({2 - \sqrt{3})}^{2} }{ {2}^{2} - {(\sqrt{3})}^{2} } = \frac{ {2}^{2} + ( { \sqrt{3} )}^{2} - 4 \sqrt{7} }{4 - 3} \\ b = \frac{4 + 3 - 4 \sqrt{3} }{1} = 7 - 4 \sqrt{3}$

$a + b = 7 + {4\sqrt{3}} + 7 - 4 \sqrt{3} = 7 + 7 = 14$