6. \( 8 \sqrt{24 x^{5} y^{3}} \)

To simplify \( 8 \sqrt{24 x^{5} y^{3}} \), start by simplifying the square root. The number \( 24 \) can be factored into \( 4 \times 6 \), and \( \sqrt{4} = 2 \). So, \( \sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6} \). For the variables, \( x^5 = x^{4} \cdot x = (x^2)^2 \cdot x \), so \( \sqrt{x^5} = x^2 \sqrt{x} \). Similarly, \( y^3 = y^{2} \cdot y = (y)^2 \cdot y \), giving us \( \sqrt{y^3} = y \sqrt{y} \). Now, plugging it all back in: \[ 8 \sqrt{24 x^{5} y^{3}} = 8 \times 2 \sqrt{6} \times x^2 \sqrt{x} \times y \sqrt{y} \] When we multiply these together, we get: \[ 16 x^2 y \sqrt{6xy} \] So, the final simplified expression is \( 16 x^2 y \sqrt{6xy} \).