1. `A=-1-i,B=-2+3i,C=1-3i`
Find roots(4,A)

Solution:
Here `A=-1-i,B=-2+3i,C=1-3i`



Here `n=4`

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

Here, `a=-1` and `b=-1`

`:. r=sqrt((-1)^2+(-1)^2)=sqrt(1+1)=sqrt(2)=1.4142`

`theta=atan(b/a)+180` (Since `a<0`)

`:. theta=atan((-1)/(-1))+180`

`:. theta=atan(1)+180`

`:. theta=45+180`

`:. theta=225 ^circ` or `theta=(5pi)/(4)` rad = 3.927 rad

`:. theta=3.927`

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

`=1.4142*(cos(3.927)+i*sin(3.927))`

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=1.0905*[cos((3.927+2*0*pi)/4)+i*sin((3.927+2*0*pi)/4)]`

`w_0=1.0905*[cos(0.9817)+i*sin(0.9817)]`

`w_0=0.6059+0.9067i`


for `k=1`

`w_1=1.0905*[cos((3.927+2*1*pi)/4)+i*sin((3.927+2*1*pi)/4)]`

`w_1=1.0905*[cos(2.5525)+i*sin(2.5525)]`

`w_1=-0.9067+0.6059i`


for `k=2`

`w_2=1.0905*[cos((3.927+2*2*pi)/4)+i*sin((3.927+2*2*pi)/4)]`

`w_2=1.0905*[cos(4.1233)+i*sin(4.1233)]`

`w_2=-0.6059-0.9067i`


for `k=3`

`w_3=1.0905*[cos((3.927+2*3*pi)/4)+i*sin((3.927+2*3*pi)/4)]`

`w_3=1.0905*[cos(5.6941)+i*sin(5.6941)]`

`w_3=0.9067-0.6059i`