Verified answer

[tex]\displaystyle\bf\\\frac{x^{\dfrac{1}{3} } +\sqrt[6]{xy} }{\sqrt[6]{xy} +y^{\dfrac{1}{3} } } =\frac{\Big(x^{\dfrac{1}{6}} \Big)^{2} +x^{\dfrac{1}{6} } \cdot y^{\dfrac{1}{6} } }{x^{\dfrac{1}{6} } \cdot y^{\dfrac{1}{6} }+\Big(y^{\dfrac{1}{6} }\Big)^{2} } =\frac{x^{\dfrac{1}{6}} \cdot\Big(x^{\dfrac{1}{6} } + y^{\dfrac{1}{6} }\Big) }{y^{\dfrac{1}{6} } \cdot \Big(x^{\dfrac{1}{6} }+ y^{\dfrac{1}{6} }\Big) } =\\\\\\=\frac{x^{\dfrac{1}{6} } }{y^{\dfrac{1}{6} } }[/tex]