1. `A=5+6i`
Find roots(4,A)

Solution:
Here `A=5+6i`



Here `n=4`

Step-1: Convert to polar form: `z=r*(cos(theta)+i*sin(theta))`

Here, `a=5` and `b=6`

`:. r=sqrt(5^2+6^2)=sqrt(25+36)=sqrt(61)=7.8102`

`theta=atan(b/a)` (Since `a>0`)

`:. theta=atan((6)/(5))`

`:. theta=atan(1.2)`

`:. theta=50.1944 ^circ` or `theta=0.8761` rad

`:. theta=0.8761`

The polar form is `r*(cos(theta)+i*sin(theta))`

`=7.8102*(cos(0.8761)+i*sin(0.8761))`

Step-2: All `n^(th)` roots of given complex number `r*(cos(theta)+i*sin(theta))` is

`w_k=root (n)(r)*[cos((theta+2kpi)/n)+i*sin((theta+2kpi)/n)]`, `k=0,..,n-1`

Hence the 4 roots are
for `k=0`

`w_0=1.6717*[cos((0.8761+2*0*pi)/4)+i*sin((0.8761+2*0*pi)/4)]`

`w_0=1.6717*[cos(0.219)+i*sin(0.219)]`

`w_0=1.6318+0.3632i`


for `k=1`

`w_1=1.6717*[cos((0.8761+2*1*pi)/4)+i*sin((0.8761+2*1*pi)/4)]`

`w_1=1.6717*[cos(1.7898)+i*sin(1.7898)]`

`w_1=-0.3632+1.6318i`


for `k=2`

`w_2=1.6717*[cos((0.8761+2*2*pi)/4)+i*sin((0.8761+2*2*pi)/4)]`

`w_2=1.6717*[cos(3.3606)+i*sin(3.3606)]`

`w_2=-1.6318-0.3632i`


for `k=3`

`w_3=1.6717*[cos((0.8761+2*3*pi)/4)+i*sin((0.8761+2*3*pi)/4)]`

`w_3=1.6717*[cos(4.9314)+i*sin(4.9314)]`

`w_3=0.3632-1.6318i`