Question

Освободитесь от иррациональности в знаменателе дроби:

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

Answer 1

$frac{1}{ sqrt{2}+sqrt{5}-sqrt{3}}= frac{sqrt2+sqrt5+sqrt3}{(sqrt2+sqrt5-sqrt3)(sqrt2+sqrt5+sqrt3)}= frac{sqrt2+sqrt5+sqrt3}{(sqrt2+sqrt5)^2-3}=\\=frac{sqrt2+sqrt5+sqrt3}{7+2sqrt{10}-3}=frac{sqrt2+sqrt5+sqrt3}{5+2sqrt{10}}=frac{(sqrt2+sqrt5+sqrt3)(5-2sqrt{10})}{(5+2sqrt{10})(5-2sqrt{10})}=\\= frac{(sqrt2+sqrt5+sqrt3)(5-2sqrt{10})}{25-4cdot 10} =-frac{(sqrt2+sqrt5+sqrt3)(5-2sqrt{10})}{15}$