Simplify the following expressions: (1) \( \frac{4^{x} \cdot 2^{x-2}}{8^{x-1}} \)

To simplify the expression \( \frac{4^{x} \cdot 2^{x-2}}{8^{x-1}} \), we can first express everything in terms of base 2. 1. Rewrite \( 4^{x} \) and \( 8^{x-1} \): - \( 4^{x} = (2^2)^{x} = 2^{2x} \) - \( 8^{x-1} = (2^3)^{x-1} = 2^{3(x-1)} = 2^{3x - 3} \) Now, substitute these into the original expression: \[ \frac{2^{2x} \cdot 2^{x-2}}{2^{3x-3}} \] 2. Combine the numerators: - \( 2^{2x} \cdot 2^{x-2} = 2^{2x + (x - 2)} = 2^{3x - 2} \) So, the expression simplifies to: \[ \frac{2^{3x - 2}}{2^{3x - 3}} \] 3. Use the properties of exponents: - \( \frac{a^m}{a^n} = a^{m-n} \) This gives us: \[ 2^{(3x - 2) - (3x - 3)} = 2^{3 - 2} = 2^1 = 2 \] Thus, the simplified expression is: \[ \boxed{2} \]