Ta có:
\(x=\sqrt[3]{2+\sqrt{3}}+\sqrt[3]{2-\sqrt{3}}\)
\(\Leftrightarrow x^3=2+\sqrt{3}+2-\sqrt{3}+3\left(\sqrt[3]{2+\sqrt{3}}+\sqrt[3]{2-\sqrt{3}}\right).\sqrt[3]{2+\sqrt{3}}.\sqrt[3]{2-\sqrt{3}}\)
\(\Leftrightarrow x^3=4+3x\)
\(\Leftrightarrow x^3-3x-4=0\)
Thế vào ta có:
\(M=2015\left(x^3-3x-5\right)^{2014}=2015\left[\left(x^3-3x-4\right)-1\right]^{2014}\)
\(=2015.\left(-1\right)^{2014}=2015\)
\(x=\sqrt[3]{2+\sqrt{3}}+\sqrt[3]{2-\sqrt{3}}\)
\(x^3=2+\sqrt{3}+2-\sqrt{3}+3.\sqrt[3]{2+\sqrt{3}}.\sqrt[3]{2-\sqrt{3}}\left(\sqrt[3]{2+\sqrt{3}}+\sqrt[3]{2-\sqrt{3}}\right)\)
\(x^3=4+3x\)
\(x^3-3x=4\)
\(M=2015\left(x^3-3x-5\right)^{2014}=2015\left(4-5\right)^{2014}=2015\)
