1. `A=5+6i`
Find Cube root of complex number

Solution:
Here `A=5+6i`


Step-1: Convert to exponential form: `z = re^(i 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`

Exponential form:
`5+6i=r*e^(i*theta)`

`5+6i=7.8102*e^(i(0.8761))`

Step-2: Apply the power formula
Now `(5+6i)^(0.3333)=(7.8102)^(0.3333)*e^(i(0.3333*0.8761))`

`=1.9841*e^(i(0.292))`

Step-3: Convert back to rectangular form
`=1.9841*(cos(0.292)+isin(0.292))`

`=1.9841*(0.9577+0.2879i)`

`=1.9001+0.5712i`