Đặt: \(\left\{ \begin{array}{l}u = {4^x} - 8\\v = {2^x} - 64\end{array} \right.\)
Khi đó bất phương trình trở thành:
\({u^3} + {v^3} = {\left( {u + v} \right)^3} \Leftrightarrow {u^3} + {v^3} = {u^3} + {v^3} + 3uv\left( {u + v} \right) \Leftrightarrow uv\left( {u + v} \right) = 0\)
\(\begin{array}{l} \Leftrightarrow \left[ \begin{array}{l}u = 0\\v = 0\\u + v = 0\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}{4^x} - 8 = 0\\{2^x} - 64 = 0\\{4^x} + {2^x} - 72\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}{4^x} = 8\\{2^x} = 64\\{2^x} = 8\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \frac{3}{2}\\x = 6\\x = 3\end{array} \right.\\ \Rightarrow \left[ \begin{array}{l}{x_1} = \frac{3}{2}\\{x_2} = 6\\{x_3} = 3\end{array} \right. \Rightarrow {x_1} + {x_2} + {x_3} = \frac{{21}}{2}\end{array}\)
