Question

If ( x + iy ) 3 = u + iv , then show that .

Answer 1

The given expression is,

( x+iy ) 3 =u+iv

Using the identity of ( a+b ) 3

x 3 + ( iy ) 3 +3⋅x⋅iy( x+iy )=u+iv x 3 + i 3 y 3 +3 x 2 yi+3x y 2 i 2 =u+iv x 3 −i y 3 +3 x 2 yi−3x y 2 =u+iv ( x 3 −3x y 2 )+i( 3 x 2 y− y 3 )=u+iv

Comparing real and imaginary parts, we get

u= x 3 −3x y 2 , v=3 x 2 y− y 3

u x + v y = x 3 −3x y 2 x + 3 x 2 y− y 3 y = x( x 2 −3 y 2 ) x + y( 3 x 2 − y 2 ) y = x 2 −3 y 2 +3 x 2 − y 2 =4( x 2 − y 2 )

Hence, the expression has been proved.