Question

Simplify 1/( 2+√3) + 2/(√5- √3) + 1/(2-√5)

Answer 1

Answer:

$\huge{\boxed{\sf{ \frac{1}{2 + \sqrt{3} } + \frac{2}{ \sqrt{5} - \sqrt{3} } + \frac{1}{2 - \sqrt{5} }=0 }}}$

Step-by-step explanation:

$\huge{\boxed{\sf{SOLUTION-}}}$

$\frac{1}{2 + \sqrt{3} } + \frac{2}{ \sqrt{5} - \sqrt{3} } + \frac{1}{2 - \sqrt{5} } \\ so \: we \: rationlise \: the \: denominator \\ = \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } + \frac{2}{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5 }+ \sqrt{3} }{ \sqrt{5} + \sqrt{3} } + \frac{1}{2 - \sqrt{5} } \times \frac{2 + \sqrt{5} }{2 + \sqrt{5} } \\ = \frac{2 - \sqrt{3} }{ {2}^{2} - { \sqrt{3} }^{2} } + \frac{2 \sqrt{5} + 2 \sqrt{3} }{ { \sqrt{5} }^{2} - { \sqrt{3} }^{2} } + \frac{2 + \sqrt{5} }{ {2}^{2} - { \sqrt{5} }^{2} } \\ = \frac{2 - \sqrt{3} }{4 - 3} + \frac{2 \sqrt{5} + 2 \sqrt{3} }{5 - 3} + \frac{2 + \sqrt{5} }{4 - 5} \\ \frac{2 - \sqrt{3} }{1} + \frac{2 \sqrt{5} + 2 \sqrt{3} }{2} - \frac{2 + \sqrt{5} }{1} \\ = \frac{4 - 2 \sqrt{3} + 2 \sqrt{5} + 2 \sqrt{3} - (4 + 2 \sqrt{5} ) }{2} \\ = \frac{4 - 4- 2 \sqrt{3} + 2 \sqrt{3} + 2 \sqrt{5} - 2 \sqrt{5} }{2} \\ = \frac{0}{2} \\ = 0$

$\huge{\boxed{\sf{ \frac{1}{2 + \sqrt{3} } + \frac{2}{ \sqrt{5} - \sqrt{3} } + \frac{1}{2 - \sqrt{5} }=0 }}}$