<p>{ 2xy ^-2 y ^4 / 3x ^-1 y } ^ -2 times { 4xy / 2x^-1 y^-3 } ^2</p>
<p>If you have variable^(-power) in the numerator of a fraction, you can replace it with variable^power in the denominator, and vice versa. The problem becomes</p>
<p>{ 2x y^4 x^1 / 3 y y^2 } ^ -2 times { 4xy x^1 y^3 / 2 } ^ 2
= {2 x^2 y^4 / 3 y^3} ^ -2 times { 4x^2 y^4 /2 } ^ 2
= {2 x^2 y / 3} ^ -2 times { 2 x^2 y^4} ^ 2
= { 3 / 2x^2 y } ^ 2 times {2 x^2 y^4 } ^ 2
= { (3 / 2x^2 y) (2 x^2 y^4) } ^ 2
= { 3 y^3 } ^ 2
= 9 y^6 (barring any math error along the way :) )</p>