Question

Tìm \(x\in N\) sao cho \(\left(2^x-8\right)^3+\left(4^x+13\right)^3=\left(4^x+2^x+5\right)^3\)

Answer 1

(2^x - 8 + 4^x + 13) [(2^x - 8)² + (2^x-8)(4^x + 13) + (4^x + 13)²] = (4^x + 2^x +5)³

\(\Leftrightarrow\)(2^x + 4^x + 5) [(2^x - 8)² + (2^x-8)(4^x + 13) + (4^x + 13)²] = (4^x + 2^x +5)³

\(\Leftrightarrow\)(2^x - 8)² + (2^x-8)(4^x + 13) + (4^x + 13)²= (4^x + 2^x +5)²

\(\Leftrightarrow\)(2^x -8 + 4^x + 13)² - (2^x -8)(4^x + 13) = (4^x + 2^x +5)²

\(\Leftrightarrow\)(4^x +2^x + 5)² - (2^x -8)(4^x +13) = (4^x + 2^x +5)²

\(\Leftrightarrow\) (2^x - 8) (4^x + 13) = 0

\(\Leftrightarrow\left[{}\begin{matrix}2^x-8=0\\4^x+13=0\end{matrix}\right.\Leftrightarrow}x=4\)

Vậy x = 4.