\(S^3=\left(\sqrt[3]{5\sqrt{2}+7}\right)^3-3.\left(\sqrt[3]{5\sqrt{2}+7}\right)^2.\sqrt[3]{5\sqrt{2}+7}+3\left(\sqrt[3]{5\sqrt{2}-7}\right)^2.\sqrt[3]{5\sqrt{2}-7}-\left(\sqrt[3]{5\sqrt{2}-7}\right)^3\)
\(S^3=5\sqrt{2}+7-3.\sqrt[3]{5\sqrt{2}+7}.\sqrt[3]{5\sqrt{2}-7}.\left(\sqrt[3]{5\sqrt{2}+7}-\sqrt[3]{5\sqrt{2}-7}\right)-\left(5\sqrt{2}-7\right)\)
\(S^3=5\sqrt{2}+7-3.\sqrt[3]{\left(5\sqrt{2}+7\right).\left(5\sqrt{2}-7\right)}.S-5\sqrt{2}+7\)
\(S^3=14-3.\sqrt[3]{50-49}.S=14-3S\)
\(\Rightarrow S^3+3S-14=0\Rightarrow S^3-2S^2+2S^2-4S+7S-14=0\)
=> S2(S - 2) + 2S.(S -2) + 7.(S - 2) = 0
=> (S-2).(S2 + 2S +7) = 0 => S - 2 = 0 hoăc S2 + 2S +7 = 0
+) S - 2 = 0 => S = 2.
+) S2 + 2S + 7 = 0 => vô ngiêm vì \(\Delta=-6
