Complex number division (a+ib) / (c+id) can be expressed as re^im.?

Using Euler's identity ONLY, how do I express the complex division in form re^im. Any ideas?

Complex number division (a+ib) / (c+id) can be expressed as re^im.?
(a+ib) / (c+id)


= (a+ib) (c-id)/ (c^2+d^2)


= (ac+bd)/(c^2+d^2) - [(bc-ad)/(c^2+d^2)]i


= Re + Im i, where Re = (ac+bd)/(c^2+d^2), and Im = -(bc-ad)/(c^2+d^2)





r = sqrt(Re^2+Im^2)


m = arctan(Im/Re)


(a+ib) / (c+id) = re^im


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Ideas: Find real part and imaginary part first.