\(x=\dfrac{2}{2\sqrt[3]{2}+2+\sqrt[3]{4}}\)
\(=\dfrac{2}{\sqrt[3]{16}+\sqrt[3]{8}+\sqrt[3]{4}}\)
\(=\dfrac{2}{\sqrt[3]{4}\left(1+\sqrt[3]{2}+\sqrt[3]{4}\right)}\)
\(=\dfrac{\sqrt[3]{2}\left(\sqrt[3]{2}-1\right)}{\left(1+\sqrt[3]{2}+\sqrt[3]{4}\right)\left(\sqrt[3]{2}-1\right)}\)
\(=\dfrac{\sqrt[3]{2}\left(\sqrt[3]{2}-1\right)}{\left(\sqrt[3]{2}\right)^3-1^3}=\sqrt[3]{2}\left(\sqrt[3]{2}-1\right)\)
\(y=\dfrac{6}{2\sqrt[3]{2}-2+\sqrt[3]{4}}\)
\(=\dfrac{6}{\sqrt[3]{4}\left(1-\sqrt[3]{2}+\sqrt[3]{4}\right)}\)
\(=\dfrac{3\sqrt[3]{2}.\left(\sqrt[3]{2}+1\right)}{\left(\sqrt[3]{2}\right)^3+1}=\sqrt[3]{2}.\left(\sqrt[3]{2}+1\right)\)
\(A=xy^3-x^3y\)
\(=xy\left(y^2-x^2\right)=xy\left(y-x\right)\left(y+x\right)\)
\(=\sqrt[3]{2}\left(\sqrt[3]{2}-1\right).\sqrt[3]{2}.\left(\sqrt[3]{2}+1\right)\left[\sqrt[3]{2}\left(\sqrt[3]{2}+1\right)-\sqrt[3]{2}\left(\sqrt[3]{2}-1\right)\right]\left[\sqrt[3]{2}\left(\sqrt[3]{2}-1\right)+\sqrt[3]{2}\left(\sqrt[3]{2}-1\right)\right]\)
\(=\sqrt[3]{2}\left(\sqrt[3]{2}-1\right).\sqrt[3]{2}.\left(\sqrt[3]{2}+1\right).\sqrt[3]{2}.\left(\sqrt[3]{2}+1-\sqrt[3]{2}+1\right).\sqrt[3]{2}.\left(\sqrt[3]{2}+1+\sqrt[3]{2}-1\right)\)
\(=\sqrt[3]{2}\left(\sqrt[3]{2}-1\right).\sqrt[3]{2}.\left(\sqrt[3]{2}+1\right).\left(2.\sqrt[3]{2}\right).\left(2.\sqrt[3]{2}.\sqrt[3]{2}\right)\)
\(=2.2.\left(\sqrt[3]{2}\right)^5.\left(\sqrt[3]{2}-1\right).\left(\sqrt[3]{2}+1\right)\)
\(=8.\sqrt[3]{4}.\left[\left(\sqrt[3]{2}\right)^2-1\right]=8.\sqrt[3]{4}\left(\sqrt[3]{4}-1\right)=8.\left(2\sqrt[3]{2}-\sqrt[3]{4}\right)=16\sqrt[3]{2}-8\sqrt[3]{4}\)
