Question

Cho \(x=\dfrac{1}{3}\left(1+\sqrt[3]{\dfrac{12+\sqrt{135}}{3}}+\sqrt[3]{\dfrac{12-\sqrt{135}}{3}}\right)\).

Không dùng máy tính cầm tay, hãy tính giá trị của biểu thức \(M=\left(9x^3-9x^2-3\right)^2\)

Answer 1

\(x=\dfrac{1}{3}\left(1+\sqrt[3]{\dfrac{12+\sqrt{135}}{3}}+\sqrt[3]{\dfrac{12-\sqrt{135}}{3}}\right)\)

\(\Rightarrow3x-1=\sqrt[3]{\dfrac{12+\sqrt{135}}{3}}+\sqrt[3]{\dfrac{12-\sqrt{135}}{3}}\)

\(\Rightarrow\left(3x-1\right)^3=\dfrac{12+\sqrt{135}+12-\sqrt{135}}{3}+3\sqrt[3]{\dfrac{\left(12+\sqrt{135}\right)\left(12-\sqrt{135}\right)}{9}}\left(3x-1\right)\)

\(\Rightarrow27x^3-27x^2+9x-1=8+3\left(3x-1\right)\)

\(\Rightarrow3\left(9x^3-9x^2-3\right)=-3\)

\(\Rightarrow M=\left(9x^3-9x^2-3\right)^2=1\)