Simplify the expression: \( \left(\frac{-4 r^{5} s^{3}}{r}\right)^{3} \) \( 64 r^{12} s^{9} \) \( -64 r^{4} s^{6} \) \( -4 r^{12} s^{9} \) \( -64 r^{12} s^{9} \)

To simplify the expression \( \left(\frac{-4 r^{5} s^{3}}{r}\right)^{3} \), we first simplify the fraction inside the parentheses: \[ \frac{-4 r^{5} s^{3}}{r} = -4 r^{4} s^{3} \] Now we raise this expression to the power of 3: \[ \left(-4 r^{4} s^{3}\right)^{3} = (-4)^{3} (r^{4})^{3} (s^{3})^{3} \] Calculating each part, we find: \[ (-4)^{3} = -64, \quad (r^{4})^{3} = r^{12}, \quad (s^{3})^{3} = s^{9} \] Putting it all together: \[ -64 r^{12} s^{9} \] So, the simplified expression is: \(-64 r^{12} s^{9}\) It's the correct option from the choices given!