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2015-10-08T21:57:03+05:30

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Given complex number is = 3  + 4 i.
The general way of obtaining roots for complex numbers is to convert the number in polar coordinates in the complex number space.  then obtain the root and then convert back in to Cartesian space type of representation.

    Let Cos A = 3/5  and Sin A  = 4/5   =>  A = 53.13°

Let\ Z=3+4i=5[ \frac{3}{5} + i\ \frac{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)}\\\\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.

z_0=5^{\frac{1}{7}}*[Cos\ 7.59^0+i\ Sin\ 7.59^0]\\z_1=5^{\frac{1}{7}}*[Cos\ 59.019^0+i\ Sin\ 59.019^0]\\z_2=5^{\frac{1}{7}}*[Cos\ 110.45^0+i\ Sin\ 110.45^0]\\z_3=5^{\frac{1}{7}}*[Cos\ 161.87^0+i\ Sin\ 161.87^0]\\and\ so\ on...

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