Q. 164.4( 54 Votes )
If (x + iy)3 = u + iv, then show that: 
Answer :
It is given in the question that,
(x + iy)3 = u + iv
we know that, (x + y)3 = x3 + y3 + 3xy(x + y)x3 + i3y3 + 3 × x × iy (x + iy) = u + iv
x3 + i3y3 + 3x2yi + 3xi2y2= u + iv
Now we know that, i3 = - i and i2 = -1x3 – iy3 + 3x2yi – 3xy2 = u + iv
(x3 – 3xy2) + i(-y3 + 3x2y)= u + iv
Now, equating the imaginary and real part we will get,
u = (x3 – 3xy2), v = (-y3 + 3x2y)
According to question,
= x2 – 3y2 + 3x2 – y2
= 4x2 – 4y2
= 4(x2 – y2)
Thus, 
Hence, proved
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