Chapter 9 - Solving Quadratic Equations - 9.1 - Properties of Radicals - Exercises - Page 488: 113

$x=-3$.

$27^{x}=3^{x-6}$ Rewriting $27$ as $3^{3}$, we have $(3^{3})^{x}=3^{x-6}$ $\implies 3^{3x}=3^{x-6}$ $\implies 3x=x-6$ $\implies 3x-x=-6$ $\implies 2x=-6$ $\implies x=\frac{-6}{2}=-3$ Check the result: $27^{-3}=3^{-3-6}=3^{-9}$ $3^{-3-6}=3^{-9}$