Question

Đặt \(x=\sqrt[3]{a+\dfrac{a+1}{3}\sqrt{\dfrac{8a-1}{3}}+}\sqrt[3]{a-\dfrac{a+1}{3}\sqrt{\dfrac{8a-1}{3}}}\)

CMR: với mọi \(a>\dfrac{1}{8}\) thì x là số nguyên dương

Answer 1

\(x=\sqrt[3]{a+\dfrac{a+1}{3}\sqrt{\dfrac{8a-1}{3}}}+\sqrt[3]{a-\dfrac{a+1}{3}\sqrt{\dfrac{8a-1}{3}}}\\ >\sqrt[3]{\dfrac{1}{8}+\dfrac{a+1}{3}\sqrt{\dfrac{8\cdot\dfrac{1}{8}-1}{3}}}+\sqrt[3]{\dfrac{1}{8}-\dfrac{a+1}{3}\sqrt{\dfrac{8\cdot\dfrac{1}{8}-1}{3}}}\\ =\sqrt[3]{\dfrac{1}{8}+\dfrac{a+1}{3}\sqrt{\dfrac{1-1}{3}}}+\sqrt[3]{\dfrac{1}{8}-\dfrac{a+1}{3}\sqrt{\dfrac{1-1}{3}}}\\ =\sqrt[3]{\dfrac{1}{8}}+\sqrt[3]{\dfrac{1}{8}}=\dfrac{1}{2}+\dfrac{1}{2}=1>0\)

Vậy................