1. `A=-1-i,B=-2+3i,C=1-3i`
Find cbrt(A)Solution:Here `A=-1-i,B=-2+3i,C=1-3i`
For a complex number `z=a+bi`, the polar form is `z=r*(cos(theta)+i*sin(theta))`
then power of n of given complex number can be obtained by
`z^n=[r*(cos(theta)+i*sin(theta))]^n=r^n*[cos(n*theta)+i*sin(n*theta)]`
Step-1: Convert to exponential form: `z = re^(i 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=1.25pi` rad = 3.927 rad
`:. theta=3.927`
Exponential form:`-1-i=r*e^(i*theta)`
`-1-i=1.4142*e^(i(3.927))`
Step-2: Apply the power formulaNow `(-1-i)^(0.3333)=(1.4142)^(0.3333)*e^(i(0.3333*3.927))`
`=1.1225*e^(i(1.309))`
Step-3: Convert back to rectangular form`=1.1225*(cos(1.309)+isin(1.309))`
`=1.1225*(0.2588+0.9659i)`
`=0.2905+1.0842i`