Question

\(\sqrt[3]{5\sqrt{2}+7}-\sqrt[3]{5\sqrt{2}-7}\)

Answer 1

Đặt \(A=\sqrt[3]{5\sqrt{2}+7}-\sqrt[3]{5\sqrt{2}-7}=\sqrt[3]{7+5\sqrt{2}}+\sqrt[3]{7-5\sqrt{2}}\)

\(\Leftrightarrow A^3=14+3\left(-1\right)A\)

\(\Leftrightarrow\left(A^3-2A^2\right)+\left(2A^2-4A\right)+\left(7A-14\right)=0\)

\(\Leftrightarrow\left(A-2\right)\left(A^2+2A+7\right)=0\)

\(\Leftrightarrow A=2\)

Answer 2

\(\sqrt[3]{5\sqrt{2}+7}-\sqrt[3]{5\sqrt{2}-7}\)

\(=\sqrt[3]{1+3\sqrt{2}+6+2\sqrt{2}}-\sqrt[3]{-1+3\sqrt{2}-6+2\sqrt{2}}\)

\(=\sqrt[3]{\left(1+\sqrt{2}\right)^3}-\sqrt[3]{\left(\sqrt{2}-1\right)^3}\)

\(=1+\sqrt{2}-\sqrt{2}+1=2\)