If (x + iy)3 = (u + iv) then prove that = 4 (x2 – y2).
Given that, (x + iy)3 = (u + iv)
⇒ x3 + (iy)3 + 3x2iy + 3xi2y2 = u + iv
⇒ x3 - iy3 + 3x2iy - 3xy2 = u + iv
⇒ x3 - 3xy2 + i(3x2y - y3) = u + iv
On equating real and imaginary parts, we get
U = x3 - 3xy2 and v = 3x2y - y3
Now ,
= x2 - 3y2 + 3x2 - y2
= 4x2 - 4y2
= 4(x2 - y2)
Hence,