Question

20 БАЛЛОВ

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

Answer 1

1) $frac{1}{sqrt{5} + sqrt{2}} + frac{1}{sqrt{5} - sqrt{2}} = frac{sqrt{5} - sqrt{2}+ sqrt{5} + sqrt{2}}{(sqrt{5} + sqrt{2})(sqrt{5}-sqrt{2})} = frac{2sqrt{5}}{5-2} = frac{2sqrt{5}}{3}$
2) $frac{sqrt{2}}{2-sqrt{2}} + frac{sqrt{2}}{2+sqrt{2}} = frac{sqrt{2}(2+sqrt{2}) + sqrt{2}(2-sqrt{2})}{(2-sqrt{2})(2+sqrt{2})}=frac{4sqrt{2}}{4-2}=2sqrt{2}$
3) $frac{2}{sqrt{2}+1}-frac{2}{sqrt{2}-1} + frac{3}{2sqrt{2}}=frac{2(sqrt{2}-1) - 2(sqrt{2}+1)}{(sqrt{2}+1)(sqrt{2}-1)} + frac{3}{2sqrt{2}}=frac{-4}{1} + frac{3}{2sqrt{2}}=frac{3-8sqrt{2}}{2sqrt{2}}$
4) $frac{3}{sqrt{3}+1}-frac{2}{1-sqrt{3}} + frac{3}{3sqrt{3}}=frac{3}{sqrt{3}+1}+frac{2}{sqrt{3}-1} + frac{1}{sqrt{3}}=$
$=frac{3(sqrt{3}-1)+2(sqrt{3}+1)}{(sqrt{3}+1)(sqrt{3}-1)} + frac{1}{sqrt{3}}=frac{5sqrt{3}-1}{2}+frac{1}{sqrt{3}} = frac{15-sqrt{3}}{2sqrt{3}}$