Question

√5+√3/√5-√3+√5-√3/√5+√3=?

Answer 1

QUESTION

$\frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5 - \sqrt{3} } } + \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} }$

FORMULAS USED

(a - b)² = a² - 2ab - b²

(a + b) (a - b) = a² - b²

${\mathsf{\underline{\underline\purple{ANSWER}}}}$

$\frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5 } - \sqrt{3} } + \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} }$

Rationalize the denominators of the fractions

$( \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5 }- \sqrt{3} }{ \sqrt{5 } - \sqrt{ 3} } ) + ( \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \times \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} - \sqrt{3} } )$

$[\frac{ ({ \sqrt{5}) }^{2} - ( { \sqrt{3} })^{2} }{ {( \sqrt{5} - \sqrt{3}) }^{2} }]+[ \frac{ {( \sqrt{5} - \sqrt{3}) }^{2} }{( { \sqrt{5} )}^{2} - {( \sqrt{3} )}^{2} } ]$

Then we get

$[ \frac{5 - 3}{5 - 2 \sqrt{3} + 3 } ] +[ \frac{5 - 2 \sqrt{3} + 3 }{5 - 3} ]$

$\frac{2}{8 - 2 \sqrt{3} } + \frac{8 - 2 \sqrt{3} }{2}$

Take 2 as common

$\frac{2}{2(4 - \sqrt{3}) } + \frac{2(4 - \sqrt{3}) }{2}$

$\frac{1}{4 - \sqrt{3} } + 4 - \sqrt{3}$

$\frac{1 + {(4 - \sqrt{3}) }^{2} }{4 - \sqrt{3} }$

$\frac{1 + {4}^{2} - 2(4)( \sqrt{3}) { + ( \sqrt{3} )}^{2} }{4 - \sqrt{3} }$

$\frac{20 - 8 \sqrt{3} }{4 - \sqrt{3} } = \frac{4(5 - 2 \sqrt{3}) }{4 - \sqrt{3} }$

CONCEPTS USED:

→» Rationalization of surds

→» Surds