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[latex]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}[/latex][tex[latex]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})}=[/latex][/latex][latex]==frac{(a^{frac23}-b^{frac23})((a^{frac23})^3+(b^{frac23})^3)}{a^{frac23}-b^{frac23}}=a^2+b^2[/latex][latex]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}[/latex]