Đặt \(P=\sqrt[3]{1+\frac{\sqrt{84}}{9}}+\sqrt[3]{1-\frac{\sqrt{84}}{9}}\)
\(P^3=1+\frac{\sqrt{84}}{9}+1-\frac{\sqrt{84}}{9}+3\sqrt[3]{\left(1+\frac{\sqrt{84}}{9}\right)\left(1-\frac{\sqrt{84}}{9}\right)}\cdot P\)
\(P^3=2+3\sqrt[3]{1-\frac{84}{81}}\cdot P\)
\(P^3=2+3\sqrt[3]{\frac{-1}{27}}\cdot P\)
\(P^3=2+3\cdot\frac{-1}{3}\cdot P\)
\(P^3=2-P\)
\(\Leftrightarrow P^3+P-2=0\)
\(\Leftrightarrow P^3-P^2+P^2-P+2P-2=0\)
\(\Leftrightarrow P^2\left(P-1\right)+P\left(P-1\right)+2\left(P-1\right)=0\)
\(\Leftrightarrow\left(P-1\right)\left(P^2+P+2\right)=0\)
Do \(P^2+P+2>0\forall P\)
Do đó \(P-1=0\Leftrightarrow P=1\)
Vậy \(P=1\) là một số nguyên ( đpcm )
