Đặt \(A=\sqrt[3]{22\sqrt{2}+25}-\sqrt[3]{22\sqrt{2}-25}\)
\(\Rightarrow A^3=22\sqrt{2}+25-\left(22\sqrt{2}-25\right)-3\sqrt[3]{\left(22\sqrt{2}+25\right)\left(22\sqrt{2}-25\right)}.\)
\(\left(\sqrt[3]{22\sqrt{2}+25}-\sqrt[3]{22\sqrt{2}-25}\right)\)
\(=50-3\sqrt[3]{\left(22\sqrt{2}\right)^2-25^2}.A\)
\(\Rightarrow A^3=50-3A\sqrt[3]{343}\Leftrightarrow A^3=50-21A\)
\(\Leftrightarrow A^3+21A-50=0\Leftrightarrow A^3-4A+25A-50=0\)
\(\Leftrightarrow A\left(A^2-4\right)+25\left(A-2\right)=0\Leftrightarrow\left(A-2\right)\left(A+2\right)A+25\left(A-2\right)=0\)
\(\Leftrightarrow\left(A-2\right)\left(A^2+2A+25\right)=0\)
Vì \(A^2+2A+25=\left(A+1\right)^2+24>0,\forall A\Rightarrow A-2=0\Leftrightarrow A=2\)
