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Answer 1

$1)sqrt{2+sqrt{3}}=sqrt{frac{2*(2+sqrt{3})}{2}}=sqrt{frac{4+2sqrt{3}}{2}}=sqrt{frac{(sqrt{3}+1)^{2}}{2}}=frac{sqrt{3}+1}{sqrt{2}}$

Аналогично :

$2)sqrt{2-sqrt{3}}=frac{sqrt{3}-1}{sqrt{2}}$

$3)frac{2+sqrt{3}}{sqrt{2}+frac{sqrt{3}+1}{sqrt{2}}}}+frac{2-sqrt{3}}{sqrt{2}-frac{sqrt{3}-1}{sqrt{2}}}=frac{sqrt{2}*(2+sqrt{3})}{2+sqrt{3}+1}+frac{sqrt{2}*(2-sqrt{3})}{2-sqrt{3}+1}=frac{2sqrt{2}+sqrt{6}}{3+sqrt{3}}+frac{2sqrt{2}-sqrt{6}}{3-sqrt{3}}=$

$=frac{2sqrt{6}-2sqrt{2}+sqrt{18}-sqrt{6}+2sqrt{6}+2sqrt{2}-sqrt{18}-sqrt{6}}{3^{2}-(sqrt{3})^{2}}=frac{2sqrt{6}}{6}=frac{sqrt{6}}{3}$