(6) \( \frac{\sqrt{5^{4 x-2}} \cdot 125^{x+1}}{5^{x} \cdot \sqrt[3]{25^{6 x+3}}} \)

To simplify the expression \( \frac{\sqrt{5^{4 x-2}} \cdot 125^{x+1}}{5^{x} \cdot \sqrt[3]{25^{6 x+3}}} \), we can start by rewriting everything in terms of powers of 5. 1. First, notice that \( 125 = 5^3 \) and \( 25 = 5^2 \). Thus, we can rewrite \( 125^{x+1} \) and \( \sqrt[3]{25^{6x+3}} \): - \( 125^{x+1} = (5^3)^{x+1} = 5^{3(x+1)} = 5^{3x + 3} \) - \( \sqrt[3]{25^{6x+3}} = \sqrt[3]{(5^2)^{6x+3}} = \sqrt[3]{5^{12x + 6}} = 5^{\frac{12x + 6}{3}} = 5^{4x + 2} \) 2. Now we can rewrite the entire expression: \[ \frac{\sqrt{5^{4x-2}} \cdot 5^{3x + 3}}{5^x \cdot 5^{4x + 2}} \] 3. Simplifying \( \sqrt{5^{4x-2}} \) gives: \[ \sqrt{5^{4x-2}} = 5^{\frac{4x - 2}{2}} = 5^{2x - 1} \] 4. Substitute back into the expression: \[ \frac{5^{2x - 1} \cdot 5^{3x + 3}}{5^x \cdot 5^{4x + 2}} \] 5. Combine the exponents in the numerator: \[ 5^{(2x - 1) + (3x + 3)} = 5^{5x + 2} \] 6. The denominator becomes: \[ 5^{x + (4x + 2)} = 5^{5x + 2} \] 7. Therefore, the expression simplifies to: \[ \frac{5^{5x + 2}}{5^{5x + 2}} = 1 \] So, the final result is: \[ \boxed{1} \]