[tex]S=\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}\Rightarrow S^3=(\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}})^3=7+5\sqrt{2}+7-5\sqrt{2}+3.\sqrt[3]{(7+5\sqrt{2})(7-5\sqrt{2})}(\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}})=14+3(-1)S\Rightarrow S^3+3S-14=0\Rightarrow (S-2)(S^2+2S+7)=0\Rightarrow S=2[/tex]