Given \( x>0 \) and \( y>0 \), select the expression that is equivalent to \[ \sqrt[3]{-64 x^{9} y^{2}} \] Answer \( \begin{array}{ll}\text { ( } 4 i x^{3} y^{\frac{2}{3}} & -4 x^{\frac{1}{3}} y^{\frac{3}{2}}\end{array} \) Nth

First, let’s simplify the expression \( \sqrt[3]{-64 x^{9} y^{2}} \). The cube root of a negative number is expressed as \( - \) times the cube root of the positive version. Thus, we have: \[ \sqrt[3]{-64 x^{9} y^{2}} = -\sqrt[3]{64} \cdot \sqrt[3]{x^{9}} \cdot \sqrt[3]{y^{2}} = -4 x^{3} y^{\frac{2}{3}}. \] Now, looking closely, \( 4i x^{3} y^{\frac{2}{3}} \) is incorrect because it introduces an imaginary unit \( i \) which does not apply here; the cube root of a negative real number results in a negative real value, so the simplified expression is indeed: \(-4 x^{3} y^{\frac{2}{3}}.\) This ensures that the expression maintains its negative aspect without introducing imaginary numbers. So, the equivalent expression is: \(-4 x^{3} y^{\frac{2}{3}}.\)