Sửa \(84\rightarrow82\)
Đặt \(x=\sqrt[3]{1-\dfrac{\sqrt{82}}{9}};y=\sqrt[3]{1+\dfrac{\sqrt{82}}{9}}\)
\(xy=\sqrt[3]{\left(1-\dfrac{\sqrt{82}}{9}\right)\left(1+\dfrac{\sqrt{82}}{9}\right)}=\sqrt[3]{1-\dfrac{82}{81}}=\sqrt[3]{\dfrac{-1}{81}}=-\dfrac{1}{3}\)
\(\Rightarrow A=x+y\)
\(\Rightarrow A^3=x^3+y^3+3xy\left(x+y\right)\)
\(\Rightarrow A^3=1-\dfrac{\sqrt{80}}{9}+1+\dfrac{\sqrt{80}}{9}+3.\left(-\dfrac{1}{3}\right)A\)
\(\Rightarrow A^3+A-2=0\)
\(\Rightarrow A^3-1+A-1=0\)
\(\Rightarrow\left(A-1\right)\left(A^2+A+2\right)=0\)
\(\Rightarrow A-1=0\) Vì \(A^2+A+2=\left(A+\dfrac{1}{4}\right)^2+\dfrac{7}{4}>0\)
\(\Rightarrow A=1\in Z\)
\(\Rightarrowđpcm\)
