\(a=\sqrt{\sqrt[3]{x^6}+\sqrt[3]{x^4y^2}}+\sqrt{\sqrt[3]{y^6}+\sqrt[3]{y^4x^2}}\)
\(=\sqrt{\sqrt[3]{x^4}\left(\sqrt[3]{x^2}+\sqrt[3]{y^2}\right)}+\sqrt{\sqrt[3]{y^4}\left(\sqrt[3]{x^2}+\sqrt[3]{y^2}\right)}\)
\(=\sqrt{\sqrt[3]{x^2}+\sqrt[3]{y^2}}\left(\sqrt[3]{x^2}+\sqrt[3]{y^2}\right)\)\(\Rightarrow a=\left(\sqrt{\sqrt[3]{x^2}+\sqrt[3]{y^2}}\right)^3\)
\(\Rightarrow\sqrt[3]{a^2}=\sqrt[3]{x^2}+\sqrt[3]{y^2}\)