Question

simplify: (√5+√3/√5-√3)+(√2+√3/√2-√3)
​

Answer 1

TO SOLVE :–

$\\{ \bold{ \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } + \dfrac{ \sqrt{2} + \sqrt{3} }{ \sqrt{2} - \sqrt{3} } }}$

SOLUTION :–

• Let the function be –

$\\ \implies{ \bold{y = \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } + \dfrac{ \sqrt{2} + \sqrt{3} }{ \sqrt{2} - \sqrt{3} } }}$

• We should write this as –

$\\ \implies{ \bold{y = p + q}}$

▪︎ Now first solve 'p' –

$\\ \implies{ \bold{p =\dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3}}}}$

• Now rationalization –

$\\ \implies{ \bold{p =\dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3}} \times \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} } }}$

$\\ \implies{ \bold{p =\dfrac{ (\sqrt{5} + \sqrt{3} )^{2} }{ (\sqrt{5} )^{2} - (\sqrt{3})^{2} }}}$

$\\ \implies{ \bold{p =\dfrac{5 + 3 + 2 \sqrt{5} \sqrt{3} }{5 - 3}}}$

$\\ \implies{ \bold{p =\dfrac{8 + 2 \sqrt{5} \sqrt{3} }{2}}}$

$\\ \implies{ \bold{p =\dfrac{2(4 + \sqrt{5} \sqrt{3} )}{2}}}$

$\\ \implies \large{ \boxed{ \bold{p =4+ \sqrt{15} }}}$

▪︎ Now solving 'q' –

$\\ \implies{ \bold{q = \dfrac{ \sqrt{2} + \sqrt{3} }{ \sqrt{2} - \sqrt{3} } }}$

• Now rationalization –

$\\ \implies{ \bold{q = \dfrac{ \sqrt{2} + \sqrt{3} }{ \sqrt{2} - \sqrt{3} } \times \dfrac{ \sqrt{2} + \sqrt{3} }{ \sqrt{2} + \sqrt{3} } }}$

$\\ \implies{ \bold{q = \dfrac{ (\sqrt{2} + \sqrt{3} )^{2} }{ (\sqrt{2})^{2} - (\sqrt{3} )^{2} }}}$

$\\ \implies{ \bold{q = \dfrac{2 + 3 + 2 \sqrt{2} \sqrt{3} }{2 - 3}}}$

$\\ \implies{ \bold{q = \dfrac{5+ 2 \sqrt{2} \sqrt{3} }{ - 1}}}$

$\\ \implies \large{ \boxed{ \bold{q = - 5 - 2 \sqrt{6} }}}$

• So that –

$\\ \implies{ \bold{y =4+ \sqrt{15}- 5 - 2 \sqrt{6}}}$

$\\ \implies \large{ \boxed{{ \bold{y = \sqrt{15} - 2 \sqrt{6} - 1}}}}$

Answer 2

Step-by-step explanation:

${\red{\underline{\underline{\bold{Given:-}}}}}$

• $\frac{ \sqrt{5 } + \sqrt{3} }{ \sqrt{5} - 3 } + \frac{ \sqrt{2} + \sqrt{3} }{ \sqrt{2} - \sqrt{3} }$

${\blue{\underline{\underline{\bold{Concept\:used\:here}}}}}$

• Rationalising of the numbers.

${\green{\underline{\underline{\bold{Solution:-}}}}}$

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

______________

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

By rationalising:-

$\frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \\ \\ = \frac{{( \sqrt{5} + \sqrt{3}) }^{2}}{ {( \sqrt{5} )}^{2} - {( \sqrt{3}) }^{2} } \\ \\ = \frac{ {( \sqrt{5}) }^{2} + {( \sqrt{3} )}^{2} + 2( \sqrt{5}) (\sqrt{3} )}{5 - 3} \\ \\ = \frac{5 + 3 + 2 \sqrt{15} }{2} \\ \\ = \frac{8 + 2 \sqrt{15} }{2} \\ \\ = \frac{2(4 + \sqrt{15}) }{2} \\ \\ = 4 + \sqrt{15}$

_______________

$\frac{ \sqrt{2} + \sqrt{3} }{ \sqrt{2} - \sqrt{3} } \times \frac{ \sqrt{2} + \sqrt{3} }{ \sqrt{2} + \sqrt{3} } \\ \\ = \frac{{( \sqrt{2} + \sqrt{3}) }^{2}}{ {( \sqrt{2} )}^{2} - {( \sqrt{3}) }^{2} } \\ \\ = \frac{ {( \sqrt{2}) }^{2} + {( \sqrt{3} )}^{2} + 2( \sqrt{2}) (\sqrt{3} )}{2 - 3} \\ \\ = \frac{2 + 3 + 2 \sqrt{6} }{ - 1} \\ \\ = \frac{5 + 2 \sqrt{6} }{ - 1} \\ \\ = - 5 - 2 \sqrt{6}$

_______________

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

$4 + \sqrt{15} - 5 - 2 \sqrt{6}$

$\boxed{ = \sqrt{15} - 2 \sqrt{6} -1 }$