The complex number is
#z=-1+i#
And we need #z^(1/3)#
The polar form of #z# is
#z=|z|(costheta+isintheta)#
where,
#{(|z|=sqrt((-1)^2+(1)^2)),(costheta=-1/|z|),(sintheta=1/|z|):}#
#=>#, #{(|z|=sqrt2),(costheta=-1/sqrt2),(sintheta=1/sqrt2):}#
#=>#, #theta=3/4pi+2kpi#
Therefore,
#z=(sqrt2)^(1/3)(cos(3/4pi+2kpi)+isin(3/4pi+2kpi))#
So, By De Moivre's theorem
#z^(1/3)=(sqrt2)^(1/3)(cos(1/4pi+2/3kpi)+isin(1/4pi+2/3kpi))#
When
#k=0#, #=>#, #z_0=(sqrt2)^(1/3)(cos(1/4pi)+isin(1/4pi))#
#=(sqrt2)^(1/3)(1/sqrt2+i/sqrt2)#
#=2^(-1/3)+i2^(-1/3)#
#=0.7937+i0.7937#
#k=1#, #=>#, #z_1=(sqrt2)^(1/3)(cos(11/12pi)+isin(11/12pi))#
#=-1.084+i0.2905#
#k=2#, #=>#, #z_2=(sqrt2)^(1/3)(cos(19/12pi)+isin(19/12pi))#
#=0.2905-i1.084#
The solution is #{0.7937+i0.7937, -1.084+i0.2905, 0.2905-i1.084 }#
