Question

Retionlise the denominator 1/√3+√5+√7​

Answer 1

$\sf \frac{1}{ \sqrt{3} + \sqrt{5} + 7} \times \frac{( \sqrt{3} + \sqrt{5} - \sqrt{7}) }{ (\sqrt{3} + \sqrt{5} - \sqrt{7}) }$

$\sf \frac{ \sqrt{3} + \sqrt{5} - \sqrt{7} }{3 + 5 + 2 \sqrt{15 - 7}} = \frac{ \sqrt{3} + \sqrt{5} - \sqrt{2}}{1 + 2 \sqrt{15}} \times \frac{1 - 2 \sqrt{15}}{1 - 2 \sqrt{15}}$

$\sf \red\implies \frac{( \sqrt{3} + \sqrt{5} - \sqrt{7}) \times - 2 \sqrt{15} }{1 - 60}$

$\sf\purple\implies \frac{ - 6 \sqrt{5} - 10 \sqrt{3} + 14 \sqrt{15}}{ - 59}$

$\sf \blue\implies \frac{6 \sqrt{5} + 10 \sqrt{3} - 14 \sqrt{15} }{59}$

Answer 2

$\bf \frac{ \sqrt{3} + \sqrt{5} - \sqrt{7} }{3 + 5 + 2 \sqrt{15 - 7}} = \frac{ \sqrt{3} + \sqrt{5} - \sqrt{2}}{1 + 2 \sqrt{15}} \times \frac{1 - 2 \sqrt{15}}{1 - 2 \sqrt{15}}$

$\bf \implies\blue {\dfrac{( \sqrt{3} + \sqrt{5} - \sqrt{7}) \times - 2 \sqrt{15} }{1 - 60}}$

$\bf \implies\red {\dfrac{ - 6 \sqrt{5} - 10 \sqrt{3} + 14 \sqrt{15}}{ - 59}}$

$\bf \implies \orange {\dfrac{6 \sqrt{5} + 10 \sqrt{3} - 14 \sqrt{15} }{59}}$