Question

In Problems 53–60, find all the complex roots. Leave your answers in polar form with the argument in degrees.

55. The complex fourth roots of$4-4\sqrt{3}i$.

Short answer

The complex fourth roots of$4-4\sqrt{3}i$are$\sqrt[4]{8}(\cos 75+i\sin 75),\sqrt[4]{8}(\cos 165+i\sin 165),\sqrt[4]{8}(\cos 255+i\sin 255),\sqrt[4]{8}(\cos 345+i\sin 345)$.

Step-by-step solution

Step 1. Rewrite 4-43i in the polar form.

The magnitude of $4-4\sqrt{3}i$is

role="math" localid="1646848982162" $$\begin{gathered} \sqrt{4^2+{(-4\sqrt{3})}^2}=\sqrt{16+48} \\ =\sqrt{64} \\ =8 \end{gathered}$$

which gives

$$\begin{gathered} 4-4\sqrt{3}=8(\frac{1}{2}-\frac{\sqrt{3}}{2}i) \\ =8(\cos 300°-i\sin 300°) \end{gathered}$$

Step 2. Using the theorem of complex roots.

We get

$$\begin{gathered} z_k=\sqrt[4]{8}\left[\cos \right(\frac{300}{4}+\frac{360k}{4})+i\sin (\frac{300}{4}+\frac{360k}{4}\left)\right];k=0,1,2,3 \\ =\sqrt[4]{8}\left[\cos \right(75°+90°k)+i\sin (75°+90°k\left)\right] \end{gathered}$$

Step 3. Substitute k=0,1,2,3 in zk=84[cos(75+90k)+isin(75+90k)].

We get

$$\begin{gathered} z_0=\sqrt[4]{8}(\cos 75+i\sin 75) \\ {z}_1=\sqrt[4]{8}\left[\cos \right(75+90)+i\sin (75+90\left)\right] \\ =\sqrt[4]{8}(\cos 165+i\sin 165) \\ {z}_2=\sqrt[4]{8}\left[\cos \right(75+180)+i\sin (75+180\left)\right] \\ =\sqrt[4]{8}(\cos 255+i\sin 255) \\ {z}_3=\sqrt[4]{8}\left(\cos \right(75+270)+i\sin (75+270\left)\right) \\ =\sqrt[4]{8}(\cos 345+i\sin 345) \end{gathered}$$

So we have found the complex fourth roots of$4-4\sqrt{3}i$.