\( 18^{6}=3^{6}\times 2^{x}\times 6^{y} \)

To solve the equation \( 18^{6} = 3^{6} \times 2^{x} \times 6^{y} \), let's start by breaking down \( 18 \): \[ 18 = 2 \times 3^{2} \] Now, we can express \( 18^{6} \): \[ 18^{6} = (2 \times 3^{2})^{6} = 2^{6} \times 3^{12} \] Next, let's rewrite \( 6^{y} \): \[ 6 = 2 \times 3 \] \[ 6^{y} = (2 \times 3)^{y} = 2^{y} \times 3^{y} \] Now, substituting this into our equation gives us: \[ 2^{6} \times 3^{12} = 3^{6} \times 2^{x} \times (2^{y} \times 3^{y}) \] Simplifying the right side: \[ 2^{6} \times 3^{12} = 2^{x+y} \times 3^{6+y} \] Now we can equate the coefficients of \( 2 \) and \( 3 \): For \( 2 \): \[ 6 = x + y \] For \( 3 \): \[ 12 = 6 + y \] From the second equation, we have: \[ y = 6 \] Substituting \( y = 6 \) into the first equation: \[ 6 = x + 6 \] \[ x = 0 \] Thus, the solution is: \[ x = 0, \quad y = 6 \]