If a,b,p,q are real and #(a-ib)^(1/3)=p-iq# prove that #(a+ib)^(1/3)=p+iq#?

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Answer 1 · 2017-12-28T15:37:12

See below.

We have

#(p+i q)^3 = p^3 - 3 p q^2 + i (3 p^2 q - q^3) = a+ i b#

and

#(p-i q)^3 = p^3 - 3 p q^2 - i (3 p^2 q - q^3) = a - i b#

NOTE

Here

#a = p^3 - 3 p q^2#
#b = 3 p^2 q - q^3#