We can take any complex number in the form of a+ib ,
take nth root in the form of x+iy . Here, n is any positive integer.
so, (a+ib)n=x+iy
As for De Moivre's Theorem, complex number
z=cosϕ+isinϕ ,
z1/n=r1/n(cosα+isinα) (I)
where α=(ϕ+2πk)/n ; k=1,2,3,...,(n−1)
so, we can get the nth root of the complex number as the form of z bar:
z=r(cosϕ+isinϕ)
To take a general equation as
zˉ=a−ib
Polar coordinate form of this is zˉ=r(cosϕ+isinϕ)
Using De Moivre's theorem again, we get roots in the conjugate case
zˉ1/n=r1/n(cosα−isinα) (II)
From equations (I) and (II), we can get that real parts of the nth part of the equation
Re(zˉ1/n)=Re(z1/n) are the same.
Then proved.
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