Question

Find all the seventh roots of (3+4i).

Answer 1

$Z=x+iy=3+4i=5[ 3/5 + i\ 4/5 ]\\\\ = 5 * [CosA+ i\ SinA ]=5*[Cos(2n\pi+A)+i\ Sin(2n\pi+A) ]\\\\=5*e^{i\ A}=5*e^{i\ (2n\pi\ +A)}\\\\ where\ \ tan\ A=\frac{4}{3},\ \ A=53.13^0\\\\Z^{\frac{1}{7}}=5^{\frac{1}{7}}e^{\frac{1}{7}(2n\pi+A)i}\\\\=5^{\frac{1}{7}}*[Cos\frac{(2n\pi+A)}{7}+i\ Sin\frac{(2n\pi+A)}{7}]$

Substitute n =0, 1, 2, 3, 4, 5, and 6  to get  the 7 solutions.
let 5^{1/7} = N

Z1 = N [ Cos (7.59) + i Sin (7.59) ]
Z2 = N [ Cos (51.43+ 7.59) + i Sin (51.43+7.59)]
Z3 = N [ Cos (2*51.43+7.59) + i Sin( 2* 51.43+7.59) ]

and so on.