\(Pt\Leftrightarrow\sqrt[5]{27}x^{10}+2\sqrt[5]{27}=5x^6\)
Áp dụng bất đẳng thức AM-GM cho 5 số dương:
\(VT=\frac{\sqrt[5]{27}x^{10}}{3}+\frac{\sqrt[5]{27}x^{10}}{3}+\frac{\sqrt[5]{27}x^{10}}{3}+\sqrt[5]{27}+\sqrt[5]{27}\ge5\sqrt[5]{\frac{27x^{30}}{27}}=5x^6=VF\)
Dấu = xảy ra khi \(\frac{\sqrt[5]{27}x^{10}}{3}=\sqrt[5]{27}\Leftrightarrow x^{10}=3\Leftrightarrow\orbr{\begin{cases}x=\sqrt[10]{3}\\x=-\sqrt[10]{3}\end{cases}}\)
\(\sqrt[5]{27}x^{10}-5x^6+\sqrt[5]{864}=0\)
\(3^{\frac{3}{5}}x^{10}-5x^6+2\times3^{\frac{3}{5}}=0\)
\(3^{\frac{3}{5}}\left(x^{10}+2\right)=5x^6\)
\(x=-i\sqrt{\frac{3}{3^{\frac{4}{5}}}+\frac{\sqrt[3]{10}}{3^{\frac{4}{5}}}}\)
\(x=i\sqrt{\frac{2}{3^{\frac{4}{5}}}+\frac{\sqrt[3]{10}}{3^{\frac{4}{5}}}}\)
\(x=-\sqrt{-\frac{2}{3^{\frac{4}{5}}}+\frac{\sqrt[3]{5}\left(1-i\sqrt{3}\right)}{2^{\frac{2}{3}}3^{\frac{4}{5}}}}\)
\(x=\sqrt{-\frac{2}{3^{\frac{4}{5}}}+\frac{\sqrt[3]{5}\left(1-i\sqrt{3}\right)}{2^{\frac{2}{3}}3^{\frac{4}{5}}}}\)
\(x=-\sqrt{-\frac{2}{3^{\frac{4}{5}}}+\frac{\sqrt[3]{5}\left(1+i\sqrt{3}\right)}{2^{\frac{2}{3}}3^{\frac{4}{5}}}}\)
\(\Rightarrow x=\hept{\begin{cases}\sqrt[10]{3}\\-\sqrt[10]{3}\end{cases}}\)
