((8x^3)/(27y^6))^(1/3) is the same as ((color(green)(2^3)x^3)/(color(green)(3^3)y^6))^(1/3)
Distribute the external exponent (color(red)(1/3))into the exponents inside (for both numerator and the denominator)
((color(green)(2^3)x^3)/(color(green)(3^3)y^6))^color(red)(1/3)=(color(green)(2^(3*color(red)(1/3)))x^(3*color(red)(1/3)))/(color(green)(3^(3*color(red)(1/3)))y^(6*color(red)(1/3)))=(color(green)2x)/(color(green)3y^2)