Question

^{\(-9\cdot4^{\frac{1}{x}}-5\cdot6^{\frac{1}{x}}+4\cdot9^{\frac{1}{x}}=0\)}

Answer 1

Đk: x khác 0;

pt \(\Leftrightarrow-9\cdot2^{\frac{1}{x}}-5\cdot2^{\frac{1}{x}}\cdot3^{\frac{1}{x}}+4\cdot3^{\frac{2}{x}}=0\)

\(\Leftrightarrow4\cdot\left(3^{\frac{1}{x}}\right)^2-5\cdot2^{\frac{1}{x}}\cdot3^{\frac{1}{x}}-9\cdot2^{\frac{1}{x}}=0\)

xem pt trên là pt bậc hai ẩn 3^1/x, ta có: \(\Delta=\left(5\cdot2^{\frac{1}{x}}\right)^2-4\cdot4\cdot\left(-9\cdot2^{\frac{2}{x}}\right)=169\cdot2^{\frac{1}{x}}\)

\(3^{\frac{1}{x}}=\frac{5\cdot2^{\frac{1}{x}}-13\cdot2^{\frac{1}{x}}}{2\cdot4}=-2^{\frac{1}{x}}\) (loại)

\(3^{\frac{1}{x}}=\frac{5\cdot2^{\frac{1}{x}}+13\cdot2^{\frac{1}{x}}}{2\cdot4}=\frac{9}{4}\cdot2^{\frac{1}{x}}\Leftrightarrow3^{\frac{1}{x}-2}=2^{\frac{1}{x}-2}\Leftrightarrow\frac{1}{x}-2=0\Leftrightarrow x=\frac{1}{2}\)