2.2. Simplify the following: \( 2.2 .1 \quad 4 x^{2} y^{-4} \times\left(\frac{2 y^{3}}{x^{3}}\right)^{3} \)

To simplify the expression \( 4 x^{2} y^{-4} \times\left(\frac{2 y^{3}}{x^{3}}\right)^{3} \), start by simplifying the second part: \[ \left(\frac{2 y^{3}}{x^{3}}\right)^{3} = \frac{2^{3} y^{9}}{x^{9}} = \frac{8 y^{9}}{x^{9}}. \] Now, substitute this back into the expression: \[ 4 x^{2} y^{-4} \times \frac{8 y^{9}}{x^{9}}. \] Multiply the coefficients and combine the variables: \[ \frac{4 \times 8}{1} \cdot x^{2 - 9} \cdot y^{-4 + 9} = 32 x^{-7} y^{5}. \] Finally, the expression can be rewritten as: \[ \frac{32 y^{5}}{x^{7}}. \] So, the simplified expression is: \[ \frac{32 y^{5}}{x^{7}}. \]