De Moivres formel - komplext tal med 16 lösningar

Sitter med ett komplext tal som jag har problem att få till på rätt sätt.

Lös ekvationen: ${\left(z-4-4i\right)}^{16}=2\sqrt{3}+2i$

• Variabelbyte och skriver på polär form:

$$\begin{gathered} \left[w=\left(z-4-4i\right)\right]\Rightarrow w^{16}=2\sqrt{3}+2i \\ arg w=v \iff \tan v=\frac{2}{2\sqrt{3}}=\frac{1}{\sqrt{3}}\Rightarrow v=\frac{\mathrm{\pi }}{3} \\ \left|w\right|=\sqrt{{\left(2\sqrt{3}\right)}^2+2^2}=\sqrt{4\times 3+4}=\sqrt{12+4}=\sqrt{16}=4 \\ {w}^{16}=4e^{i\frac{\mathrm{\pi }}{3}} \\ w={\left(4e^{i\frac{\mathrm{\pi }}{3}+2\mathrm{\pi n}}\right)}^{\frac{1}{16}}=4^{\frac{1}{16}}\times {\left(e^{i\frac{\mathrm{\pi }}{3}+2\mathrm{\pi n}}\right)}^{\frac{1}{16}}=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}\mathrm{\pi n}} \end{gathered}$$

• Löser för varje n:

$$\begin{gathered} n=0\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}0\pi }=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}} \\ n=1\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}1\pi }=4^{\frac{1}{16}}\times e^{i\frac{7\mathrm{\pi }}{48}} \\ n=2\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}2\pi }=4^{\frac{1}{16}}\times e^{i\frac{13\mathrm{\pi }}{48}} \\ n=3\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}3\pi }=4^{\frac{1}{16}}\times e^{i\frac{19\mathrm{\pi }}{48}} \\ n=4\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}4\pi }=4^{\frac{1}{16}}\times e^{i\frac{25\mathrm{\pi }}{48}} \\ n=5\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}5\pi }=4^{\frac{1}{16}}\times e^{i\frac{31\mathrm{\pi }}{48}} \\ n=6\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}6\pi }=4^{\frac{1}{16}}\times e^{i\frac{37\mathrm{\pi }}{48}} \\ n=7\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}7\pi }=4^{\frac{1}{16}}\times e^{i\frac{43\mathrm{\pi }}{48}} \\ n=8\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}8\pi }=4^{\frac{1}{16}}\times e^{i\frac{49\mathrm{\pi }}{48}} \\ n=9\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}9\pi }=4^{\frac{1}{16}}\times e^{i\frac{55\mathrm{\pi }}{48}} \\ n=10\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}10\pi }=4^{\frac{1}{16}}\times e^{i\frac{61\mathrm{\pi }}{48}} \\ n=11\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}11\pi }=4^{\frac{1}{16}}\times e^{i\frac{67\mathrm{\pi }}{48}} \\ n=12\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}12\pi }=4^{\frac{1}{16}}\times e^{i\frac{73\mathrm{\pi }}{48}} \\ n=13\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}13\pi }=4^{\frac{1}{16}}\times e^{i\frac{79\mathrm{\pi }}{48}} \\ n=14\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}14\pi }=4^{\frac{1}{16}}\times e^{i\frac{85\mathrm{\pi }}{48}} \\ n=15\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}15\pi }=4^{\frac{1}{16}}\times e^{i\frac{91\mathrm{\pi }}{48}} \\ n=16\Rightarrow w=4^{\frac{1}{16}}\times e^{i\frac{\mathrm{\pi }}{48}+\frac{1}{8}16\pi }=4^{\frac{1}{16}}\times e^{i\frac{97\mathrm{\pi }}{48}} \\ \left[\mathbf{n}\mathbf{=}\mathbf{16}\right]\iff \left[\mathbf{n}\mathbf{=}\mathbf{0}\right] \end{gathered}$$

• Byter tillbaka variabeln:

och det är här jag tror jag har problem.

$w=z-4-4i \iff z=w+4+4i$

Så varje lösning blir samma fast med:

$$\begin{gathered} 4+4i=\sqrt{32}{e}^{i\frac{\mathrm{\pi }}{4}} \\ Så tex för n=5: \\ z=4^{\frac{1}{16}}\times e^{i\frac{31\mathrm{\pi }}{48}}+\sqrt{32}{e}^{i\frac{\mathrm{\pi }}{4}} \end{gathered}$$

Är detta en korrekt lösning, eller finns det ett bättre sätt att göra det på?

Tack!