x^4 = -(1+i)x4=−(1+i)
Using de Moivre's identity
e^(iphi)=cosphi+isinphieiϕ=cosϕ+isinϕ we have
-(1+i) = -sqrt(2)(1/sqrt(2)+i/sqrt(2))=-sqrt(2)(cos phi_0+i sin phi_0)−(1+i)=−√2(1√2+i√2)=−√2(cosϕ0+isinϕ0)
with phi_0 = pi/4+2kpi, k = 0,pm1,pm2,cdotsϕ0=π4+2kπ,k=0,±1,±2,⋯
but
-sqrt(2)(cos phi_0+i sin phi_0) = sqrt(2)e^(ipi)e^(i(pi/4+2kpi)) = sqrt(2)e^(i(5/4pi+2kpi)) −√2(cosϕ0+isinϕ0)=√2eiπei(π4+2kπ)=√2ei(54π+2kπ)
(we used e^(ipi)+1=0eiπ+1=0 after Euler
https://en.wikipedia.org/wiki/Leonhard_Euler)
x^4 = sqrt(2)e^(i(5/4pi+2kpi)) x4=√2ei(54π+2kπ)
finally
x = root(8)(2)e^(i(5/16pi+k pi/2))x=8√2ei(516π+kπ2)
for k = 0,pm1,pm2,cdotsk=0,±1,±2,⋯
