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$1. \frac{x-y}{y^{\frac12}+x^{\frac12}}+\frac{x^{\frac12}-x}{x^{\frac12}} = \frac{(x^{\frac12})^2-(y^{\frac12})^2}{y^{\frac12}+x^{\frac12}}+\frac{x^{\frac12}(1-x^{\frac12})}{x^{\frac12}} =\\=\frac{(x^{\frac12}-y^{\frac12})(x^{\frac12}+y^{\frac12})}{y^{\frac12}+x^{\frac12}}+\frac{1-x^{\frac12}}{1} =x^{\frac12}-y^{\frac12}+1-x^{\frac12}=1-y^{\frac12}$

[tex $2) \ \frac{a^{\frac43}-b^{\frac43}}{a^{\frac13}-b^{\frac13}}* \frac{a^{\frac43}-a^{\frac23}b^{\frac23}+b^{\frac43}}{a^{\frac13}+b^{\frac13}}= \frac{(a^{\frac23})^2-(b^{\frac23})^2}{a^{\frac13}-b^{\frac13}}* \frac{a^{\frac43}-a^{\frac23}b^{\frac23}+b^{\frac43}}{a^{\frac13}+b^{\frac13}}=\\=\frac{(a^{\frac23}-b^{\frac23})(a^{\frac23}+b^{\frac23})*(a^{\frac43}-a^{\frac23}b^{\frac23}+b^{\frac43})}{(a^{\frac13}-b^{\frac13})(a^{\frac13}+b^{\frac13})}=$ [/tex]
$==\frac{(a^{\frac23}-b^{\frac23})((a^{\frac23})^3+(b^{\frac23})^3)}{a^{\frac23}-b^{\frac23}}=a^2+b^2$
$3) \ \frac{a^{\frac43}b+ab^{\frac43}}{a^{\frac13}+b^{\frac13}}*\frac1{a^{\frac13}b^{\frac13}} =\frac{ab(a^{\frac13}+b^{\frac13})}{a^{\frac13}+b^{\frac13}}*\frac1{a^{\frac13}b^{\frac13}} =\frac{ab}{1}*\frac1{a^{\frac13}b^{\frac13}}=\frac{ab}{a^{\frac13}b^{\frac13}} =a^{\frac23}b^{\frac23}$