1. `A=5+6i,B=-2+3i,C=1-3i`
Find pow(A,6)Solution:Here `A=5+6i,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=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 formulaNow `(5+6i)^(6)=(7.8102)^(6)*e^(i(6*0.8761))`
`=226981*e^(i(5.2563))`
Step-3: Convert back to rectangular form`=226981*(cos(5.2563)+isin(5.2563))`
`=226981*(0.5175-0.8557i)`
`=117469-194220i`
