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

To simplify the expression \( \frac{2^{x+2}-2^{x+1}}{2^{x}+2^{x+1}} \), we start by factoring the numerator and the denominator. In the numerator: \[ 2^{x+2} - 2^{x+1} = 2^{x+1}(2 - 1) = 2^{x+1} \] In the denominator: \[ 2^{x} + 2^{x+1} = 2^{x}(1 + 2) = 2^{x} \cdot 3 \] Now, substituting back into our expression, we have: \[ \frac{2^{x+1}}{2^{x} \cdot 3} \] Next, we can simplify further: \[ \frac{2^{x+1}}{2^{x} \cdot 3} = \frac{2^{x} \cdot 2}{2^{x} \cdot 3} = \frac{2}{3} \] Thus, the simplified form of the expression is: \[ \frac{2}{3} \]