\( 3 \quad \frac { 2 ^ { x } - 2 ^ { x - 2 } } { 2 ^ { x + 1 } - 2 ^ { x } } \)

To simplify the expression \( 3 \quad \frac { 2 ^ { x } - 2 ^ { x - 2 } } { 2 ^ { x + 1 } - 2 ^ { x } } \), let’s start by simplifying both the numerator and the denominator. In the numerator, we can factor out \( 2^{x-2} \): \[ 2^{x} - 2^{x-2} = 2^{x-2}(2^2 - 1) = 2^{x-2}(4 - 1) = 2^{x-2} \cdot 3 \] Now for the denominator, we can factor out \( 2^{x} \): \[ 2^{x+1} - 2^{x} = 2^{x}(2^1 - 1) = 2^{x}(2 - 1) = 2^{x} \cdot 1 = 2^{x} \] Substituting back into the fraction gives us: \[ 3 \quad \frac{2^{x-2} \cdot 3}{2^{x}} \] Now we can simplify this: \[ 3 \cdot \frac{2^{x-2} \cdot 3}{2^{x}} = 3 \cdot 3 \cdot \frac{1}{2^2} = 9 \cdot \frac{1}{4} = \frac{9}{4} \] Thus, the simplified form of the expression is: \[ \frac{9}{4} \]