\(P=\dfrac{a^{\dfrac{1}{3}}\cdot\sqrt{b}+b^{\dfrac{1}{3}}\cdot\sqrt{a}}{\sqrt[6]{a}+\sqrt[6]{b}}-\sqrt[3]{ab}\)
\(=\dfrac{a^{\dfrac{1}{3}}\cdot b^{\dfrac{1}{2}}+b^{\dfrac{1}{3}}\cdot a^{\dfrac{1}{2}}}{a^{\dfrac{1}{6}}+b^{\dfrac{1}{6}}}-a^{\dfrac{1}{3}}\cdot b^{\dfrac{1}{3}}\)
\(=\dfrac{a^{\dfrac{2}{6}}\cdot b^{\dfrac{3}{6}}+a^{\dfrac{3}{6}}\cdot b^{\dfrac{2}{6}}}{a^{\dfrac{1}{6}}+b^{\dfrac{1}{6}}}-a^{\dfrac{1}{3}}\cdot b^{\dfrac{1}{3}}\)
\(=\dfrac{a^{\dfrac{2}{6}}\cdot b^{\dfrac{2}{6}}\left(a^{\dfrac{1}{6}}+b^{\dfrac{1}{6}}\right)}{a^{\dfrac{1}{6}}+b^{\dfrac{1}{6}}}-a^{\dfrac{1}{3}}\cdot b^{\dfrac{1}{3}}\)
\(=a^{\dfrac{1}{3}}\cdot b^{\dfrac{1}{3}}-a^{\dfrac{1}{3}}\cdot b^{\dfrac{1}{3}}\)
=0
