Đặt \(A=\) \(\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}\)
\(A^3=7+5\sqrt{2}+7-5\sqrt{2}+3\sqrt[3]{\left(7+5\sqrt{2}\right)\left(7-5\sqrt{2}\right)}.A\)
\(\Leftrightarrow A^3=14-3A\Leftrightarrow A^3+3A-14=0\Leftrightarrow\left(A-2\right)\left(A^2+2A+7\right)=0\Leftrightarrow A-2=0\)
vì \(A^2+2A+7=\left(A+1\right)^2+6>0\)
\(\Leftrightarrow A=2\)
\(\Rightarrow\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}=2\)
\(=\sqrt[3]{2\sqrt{2}+6+3\sqrt{2}+1}+\sqrt[3]{-2\sqrt{2}+6-3\sqrt{2}+1}\)
\(=\sqrt[3]{\left(1+\sqrt{2}\right)^3}+\sqrt[3]{\left(1-\sqrt{2}\right)^3}\)
\(=1+\sqrt{2}+1-\sqrt{2}=2\)
