Question

Chứng minh \(E=\left(\sqrt[3]{2}+1\right)\cdot\sqrt[3]{\dfrac{\sqrt[3]{2}-1}{3}\in Z}\)

Answer 1

\(E=\left(\sqrt[3]{2}+1\right)\sqrt[3]{\dfrac{\sqrt[3]{2}-1}{3}}\Rightarrow E^3=\dfrac{\sqrt[3]{2}-1}{3}.\left(2+1+3\sqrt[3]{2}+3\sqrt[3]{2}^2\right)=\dfrac{\sqrt[3]{2}-1}{3}.3\left(1+\sqrt[3]{2}+\sqrt[3]{2}^2\right)=\left(\sqrt[3]{2}-1\right)\left(\sqrt[3]{2}\left(\sqrt[3]{2}+1\right)+1\right)=\sqrt[3]{2}\left(\sqrt[3]{2}^2-1\right)+\sqrt[3]{2}-1=2-\sqrt[3]{2}+\sqrt[3]{2}-1=1\Rightarrow E=1\in Z\)