\( \frac{2^{x+3}-2^{x+1}}{2 \cdot 2^{x}}+1 \quad \) (4) \( \sqrt{\frac{15^{x} \cdot 3^{x}}{7^{x+1} \cdot 5^{x-2}}} \)

To simplify the first expression \( \frac{2^{x+3}-2^{x+1}}{2 \cdot 2^{x}}+1 \), notice that you can factor out the common terms in the numerator. The expression simplifies to \( \frac{2^{x}(2^3 - 2)}{2 \cdot 2^{x}} + 1 = \frac{6}{2} + 1 = 3 + 1 = 4 \). For the second expression \( \sqrt{\frac{15^{x} \cdot 3^{x}}{7^{x+1} \cdot 5^{x-2}}} \), you can break down the components inside the square root by expressing 15 as \( 3 \cdot 5 \). This transforms the expression to \( \sqrt{\frac{(3 \cdot 5)^{x} \cdot 3^{x}}{7^{x+1} \cdot 5^{x-2}}} = \sqrt{\frac{3^{x+1} \cdot 5^{x}}{7^{x+1} \cdot 5^{x-2}}} = \sqrt{\frac{3^{x+1}}{7^{x+1}} \cdot 5^{2}} = \sqrt{ \frac{3^{x+1} \cdot 25}{7^{x+1}}} \). This simplifies to \( 5 \sqrt{\frac{3^{x+1}}{7^{x+1}}} \).