(a) \( \frac{3^{x} \cdot 2^{x-1}-6^{x}}{3.6^{x-1}} \)

To simplify the expression \( \frac{3^{x} \cdot 2^{x-1}-6^{x}}{3 \cdot 6^{x-1}} \), we can start by rewriting \( 6^{x} \) in terms of its prime factors. Since \( 6^{x} = (3 \cdot 2)^{x} = 3^{x} \cdot 2^{x} \), we can substitute this back into the expression. Now, the numerator becomes: \[ 3^{x} \cdot 2^{x-1} - 3^{x} \cdot 2^{x} = 3^{x} \cdot 2^{x-1}(1 - 2) \] So we can simplify it to: \[ 3^{x} \cdot 2^{x-1} \cdot (-1) = -3^{x} \cdot 2^{x-1} \] The denominator simplifies to: \[ 3 \cdot 6^{x-1} = 3 \cdot 6^{x-1} = 3 \cdot \frac{6^{x}}{6} = 3 \cdot \frac{3^{x} \cdot 2^{x}}{6} \] Thus the entire expression simplifies to: \[ \frac{-3^{x} \cdot 2^{x-1}}{3 \cdot \left(\frac{3^{x} \cdot 2^{x}}{6}\right)} \] After careful simplification, this ultimately yields: \[ \frac{-2}{3} \] Therefore, the simplified form of your expression \( \frac{3^{x} \cdot 2^{x-1}-6^{x}}{3 \cdot 6^{x-1}} \) is \[ \frac{-2}{3}. \]