Question

Simplify...step by step answer should be 5- root 6​

Answer 1

$\frac{1+\sqrt{2} }{\sqrt{5}+\sqrt{3} } +\frac{1-\sqrt{2} }{\sqrt{5}-\sqrt{3} }$

Taking LCM :

$\frac{(1+\sqrt{2})(\sqrt{5}-\sqrt{3})+(1-\sqrt{2})(\sqrt{5}+\sqrt{3}) }{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3}) } \\\\=\frac{(\sqrt{5}-\sqrt{3}+\sqrt{10}-\sqrt{6})+(\sqrt{5}+\sqrt{3}-\sqrt{10}-\sqrt{6} ) } {(\sqrt5)^2-\sqrt3)^2}\\\\=\frac{(\sqrt{5}-\sqrt{3}+\sqrt{10}-\sqrt{6})+(\sqrt{5}+\sqrt{3}-\sqrt{10}-\sqrt{6} ) } {(\sqrt5)^2-\sqrt3)^2}$

$=\frac{2(\sqrt5)-2(\sqrt{6})}{5-3} \\\\=\frac{2(\sqrt5-\sqrt6)}{2} \\\\=\sqrt5-\sqrt6$