Question

simplfy 5+√15/5-√15 + 5-√15/5+√15​

Answer 1

Solution :-

We have to solve

$\sf{ \dfrac{5 + \sqrt{15}}{ 5 - \sqrt{15}} + \dfrac{5 - \sqrt{15} }{ 5 + \sqrt{15}}}$

By rationalising denominator of

(i)

$\sf{ \dfrac{5 + \sqrt{15}}{ 5 - \sqrt{15}}}$

$= \sf{ \dfrac{5 + \sqrt{15}}{ 5 - \sqrt{15}} \times \dfrac{5 + \sqrt{15} }{ 5 + \sqrt{15}}}$

$= \sf{ \dfrac{(5 + \sqrt{15})^2}{ 5^2 - \sqrt{15}^2} }$

$= \sf{ \dfrac{(5 + \sqrt{15})^2}{ 25 - 15} }$

$= \sf{ \dfrac{(25 + 15 + 2.(5).\sqrt{15}}{ 5^2 - \sqrt{15}^2} }$

$= \sf{ \dfrac{40 +10\sqrt{15}}{10}}$

$= \sf{ 4 + \sqrt{5}}$

(ii)

$\sf{ \dfrac{5 - \sqrt{15}}{ 5 + \sqrt{15}}}$

$= \sf{ \dfrac{5 - \sqrt{15}}{ 5 + \sqrt{15}} \times \dfrac{5 - \sqrt{15} }{ 5 - \sqrt{15}}}$

$= \sf{ \dfrac{(5 - \sqrt{15})^2}{ 5^2 - \sqrt{15}^2} }$

$= \sf{ \dfrac{(5 - \sqrt{15})^2}{ 25 - 15} }$

$= \sf{ \dfrac{(25 + 15 - 2.(5).\sqrt{15}}{ 5^2 - \sqrt{15}^2} }$

$= \sf{ \dfrac{40 - 10\sqrt{15}}{10}}$

$= \sf{ 4 - \sqrt{5}}$

So

$\rightarrow \sf{ \dfrac{5 + \sqrt{15}}{ 5 - \sqrt{15}} + \dfrac{5 - \sqrt{15} }{ 5 + \sqrt{15}}}$

$= \sf { (4 + \sqrt{5} ) + ( 4 - \sqrt{5})}$

$= \sf{ 8}$

Answer 2

$\huge{\underline{\mathcal{\blue{SOLUTION :-}}}}$

$\star \underline{ \mathbb{OUR \: QUESTION}}$

$\sf{ \dfrac{5 + \sqrt{15}}{ 5 - \sqrt{15}} + \dfrac{5 - \sqrt{15} }{ 5 + \sqrt{15}}}$

$\star \underline{ \mathbb{RATIONALISING \: DENOMINATORS}}$

$\sf{ (I) \: \dfrac{5 + \sqrt{15}}{ 5 - \sqrt{15}}}$

$= \sf{ \dfrac{5 + \sqrt{15}}{ 5 - \sqrt{15}} \times \dfrac{5 + \sqrt{15} }{ 5 + \sqrt{15}}}$

$= \sf{ \dfrac{(5 + \sqrt{15})^2}{ 5^2 - \sqrt{15}^2} }$

$= \sf{ \dfrac{5^2 + \sqrt{15}^2 + 2(5)(\sqrt{15})}{25-15}}$

$= \sf{ \dfrac{25 + 15 + 2(5)(\sqrt{15})}{ 10} }$

$= \sf{ \dfrac{40 +10\sqrt{15}}{10}}$

$\boxed{= \sf{ 4 + \sqrt{5}}}$ ...(i)

And

$\sf{(II) \: \dfrac{5 - \sqrt{15}}{ 5 + \sqrt{15}}}$

$= \sf{ \dfrac{5 - \sqrt{15}}{ 5 + \sqrt{15}} \times \dfrac{5 - \sqrt{15} }{ 5 - \sqrt{15}}}$

$= \sf{ \dfrac{5^2 + \sqrt{15}^2 - 2(5)(\sqrt{15})}{25-15}}$

$= \sf{ \dfrac{25 + 15 - 2(5)(\sqrt{15})}{ 10} }$

$= \sf{ \dfrac{(25 + 15 - 2.(5).\sqrt{15}}{ 25 - 15} }$

$= \sf{ \dfrac{40 - 10\sqrt{15}}{10}}$

$\boxed{= \sf{ 4 - \sqrt{5}}}$ ....(ii)

$\underline{\mathbb{SO\: BY\: ADDING\: (i)\: AND\: (ii)} }$

$= \sf { (4 + \sqrt{5} ) + ( 4 - \sqrt{5})}$

$\boxed{= \sf{ 8}}$