(a) Let w=ρ(cosϕ+isinϕ)w=\rho(\cos\phi+i\sin\phi)w=ρ(cosϕ+isinϕ) and z=r(cosθ+isinθ)z=r(\cos\theta+i sin\theta)z=r(cosθ+isinθ) then by De Moivre's formula we have that zn=rn(cosnθ+isinnθ)z^n=r^n(\cos n\theta+i \sin n\theta)zn=rn(cosnθ+isinnθ)
zn=w⇝rn(cosnθ+isinnθ)=ρ(cosϕ+isinϕ)z^n=w \rightsquigarrow r^n(\cos n\theta+i \sin n\theta)=\rho(\cos\phi+i\sin\phi)zn=w⇝rn(cosnθ+isinnθ)=ρ(cosϕ+isinϕ)
Hence, rn=ρ, nθ=ϕ+2πkr^n=\rho, \,\, n\theta=\phi+2\pi krn=ρ,nθ=ϕ+2πk
w∈R−⇝ϕ=πw\in\mathbb R_- \rightsquigarrow \phi=\piw∈R−⇝ϕ=π
For n=6 :r=ρ1/6(cosπ+2πk6+isinπ+2πk6)n=6\colon r=\rho^{1/6} \bigg( \cos \frac{\pi+2\pi k}{6} +i\sin \frac{\pi+2\pi k}{6} \bigg)n=6:r=ρ1/6(cos6π+2πk+isin6π+2πk)
for each k∈{0,1,…,n−1=5}k\in\{0,1,\dots,n-1=5\}k∈{0,1,…,n−1=5}
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