Question

CMR A=\(\sqrt[3]{7-\sqrt{50}}+\sqrt[3]{7+\sqrt{50}}\)là số tự nhiên

Answer 1

\(A^3=14+3\sqrt[3]{\left(7-\sqrt{50}\right)\left(7+\sqrt{50}\right)}\left(\sqrt[3]{7-\sqrt{50}}+\sqrt[3]{7+\sqrt{50}}\right)\)

\(A^3=14+3\sqrt[3]{49-50}.A\)\(\Leftrightarrow\)\(A^3=14-3A\)

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

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

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

\(\Leftrightarrow\)\(A=2\) ( do \(A^2+2A+7=\left(A+1\right)^2+6>0\) )