Question

CMR:

\(x=\sqrt[3]{3+\sqrt{9+\dfrac{125}{7}}}-\sqrt[3]{-3+\sqrt{9+\dfrac{125}{7}}}\) là một số nguyên

Answer 1

Đề sai sửa lại là:

\(x=\sqrt[3]{3+\sqrt{9+\dfrac{125}{27}}}-\sqrt[3]{-3+\sqrt{9+\dfrac{125}{27}}}\)

\(\Leftrightarrow x=\sqrt[3]{3+\sqrt{9+\dfrac{125}{27}}}+\sqrt[3]{3-\sqrt{9+\dfrac{125}{27}}}\)

\(\Leftrightarrow x^3=3+\sqrt{9+\dfrac{125}{27}}+3-\sqrt{9+\dfrac{125}{27}}+3.\left(\sqrt[3]{3+\sqrt{9+\dfrac{125}{27}}}+\sqrt[3]{3-\sqrt{9+\dfrac{125}{27}}}\right)\left(\sqrt[3]{3+\sqrt{9+\dfrac{125}{27}}}.\sqrt[3]{3-\sqrt{9+\dfrac{125}{27}}}\right)\)

\(\Leftrightarrow x^3=6+3x.\left(\dfrac{-5}{3}\right)\)

\(\Leftrightarrow x^3+5x-6=0\)

\(\Leftrightarrow\left(x-1\right)\left(x^2+x+6\right)=0\)

\(\Leftrightarrow x=1\)

Vậy x là số nguyên