Simplify \( \sqrt{108 x^{5} y^{6}} \) \( 6 x^{2} y^{3} \) \( 6 x^{2} y^{3} \sqrt{3 x} \) \( 18 x^{3} y^{3} \)

To simplify \( \sqrt{108 x^{5} y^{6}} \), we start by breaking down the factors. First, recognize that \( 108 = 36 \times 3 \), which gives us \( \sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3} \). Next, simplify the variables: \[ \sqrt{x^5} = \sqrt{x^4 \cdot x} = \sqrt{x^4} \cdot \sqrt{x} = x^2\sqrt{x} \] \[ \sqrt{y^6} = \sqrt{(y^3)^2} = y^3 \] Putting it all together: \[ \sqrt{108 x^{5} y^{6}} = 6\sqrt{3} \cdot x^2\sqrt{x} \cdot y^3 = 6 x^{2} y^{3} \sqrt{3 x} \] So, \( \sqrt{108 x^{5} y^{6}} \) simplifies perfectly to \( 6 x^{2} y^{3} \sqrt{3 x} \). The other options like \( 6 x^{2} y^{3} \) and \( 18 x^{3} y^{3} \) are incorrect. Thus, the correct and simplified answer is \( 6 x^{2} y^{3} \sqrt{3 x} \).