Find all the sixth roots of (12+5i).?

Question

Find all the sixth roots of (12+5i).

Answer 1

Start by rewriting the number 12 + 5i in polar form. sqrt(12^2 + 5^2) = 13 and arctan(5/12) = 0.39 (radians). So 12 + 5i = 13*e^[0.39i]. The sixth roots, of which there will be 6, are equal to (13^(1/6))*e^[(0.39+2n*pi)i/6], where n assumes the values 0, 1, 2, 3, 4, and 5. The values of 0.39 + 2n*pi are equivalent to the original angle, because you add a full circle (2*pi) with each incrementation of n.

Answer 2

12+5i= 13<0.395 rad+2kpi
so the sixth root is
13^1/6 < 0.395/6 +kpi/3 with k from 0 to 5 (in polar form)
In binomial
13^1/6[( cos(0.395+kpi/3)+i sin(0.395+kpi/3)]

Answer 3

you can use combinations for this question...will be easier