\(A=\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}=\sqrt[3]{2\sqrt{2}+3.2.2+3.\sqrt{2}.4+8}+\sqrt[3]{8-3.\sqrt{2}.4+3.2.2-2\sqrt{2}}=\sqrt[3]{\left(\sqrt{2}+2\right)^3}+\sqrt[3]{\left(2-\sqrt{2}\right)^3}=\sqrt{2}+2+2-\sqrt{2}=4=\sqrt{16}\) \(B=2\sqrt{5}=\sqrt{20}\)
⇒ \(A< B\)
\(A=\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}\)
\(A^3=20+14\sqrt{2}+20-14\sqrt{2}+3\left(\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}\right).\sqrt[3]{\left(20+14\sqrt{2}\right)\left(20-14\sqrt{2}\right)}\)\(A^3=40+6A\)
\(\Leftrightarrow A^3-6A-40=0\)
\(\Leftrightarrow A^3-4A^2+4A^2-16A+10A-40=0\)
\(\Leftrightarrow A^2\left(A-4\right)+4A\left(A-4\right)+10\left(A-4\right)=0\)
\(\Leftrightarrow\left(A-4\right)\left(A^2+4A+10\right)=0\)
Do: \(A^2+4A+10=\left(A+2\right)^2+6\)
\(\Leftrightarrow A=4=\sqrt{16}< B=2\sqrt{5}=\sqrt{20}\)
\(\Rightarrow A< B\)
