Question

\(\sqrt[3]{1+\dfrac{\sqrt{84}}{9}}+\sqrt[3]{1-\dfrac{\sqrt{84}}{9}}\)

chứng minh biểu thức trên là số nguyên

Answer 1

Đặt \(A=\sqrt[3]{1+\dfrac{\sqrt{84}}{9}}+\sqrt[3]{1-\dfrac{\sqrt{84}}{9}}\)

\(\Rightarrow A^3=1+\dfrac{\sqrt{84}}{9}+1-\dfrac{\sqrt{84}}{9}+3A.\sqrt[3]{1+\dfrac{\sqrt{84}}{9}}.\sqrt[3]{1-\dfrac{\sqrt{84}}{9}}\)

\(\Leftrightarrow A^3=2+3A.\sqrt[3]{1-\dfrac{84}{81}}\)

\(\Leftrightarrow A^3=2+3A.\sqrt[3]{-\dfrac{3}{81}}=2+3A.\sqrt[3]{-\dfrac{1}{27}}\)

\(\Leftrightarrow A^3=2-A\)

\(\Leftrightarrow A^3+A-2=0\)

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

Dể thấy \(A^2+A+2=\left(A+\dfrac{1}{2}\right)^2+\dfrac{7}{4}>0\)

\(\Rightarrow A-1=0\Leftrightarrow A=1\)

Vậy \(\sqrt[3]{1+\dfrac{\sqrt{84}}{9}}+\sqrt[3]{1-\dfrac{\sqrt{84}}{9}}\) là số nguyên (đpcm)