(x + iy)^1/3 = u + iv,then prove that x/u +y/v = 4(u^2 - v^2)
(x+iy)1/3 = u+iv
cubing both sides
x+iy = (u+iv)3
= u3 +(iv)3 + 3u2 iv + 3u (iv)2
= u3 - i v3 + 3u2 v i - 3uv2
x+ iy = u3 -3uv2 + i (3u2 v - v3 )
comparing real & imaginary part
x = u3 - 3uv2
x = u(u2 -3v2 )
x/u = u2 -3v2 -----(1)
similarly y/v =(3u2 - v2 ) --->(2)
adding (1)& (2)
x/u +y/v = (u2 -3v2 ) + (3u2 - v2 )
= 4u2 - 4v2
x/u +y/v =4(u2 - v2 ) proved