Question

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

60. The complex fifth roots of$-i$.

Short answer

The complex fifth roots of$-i$are$\cos 54+i\sin 54,\cos 126+i\sin 126,\cos 198+i\sin 198,\cos 270+i\sin 270,\cos 342+\sin 342$.

Step-by-step solution

Step 1. Rewrite -i in the polar form.

The magnitude of $-i$is

$$\begin{gathered} \sqrt{0+{(-1)}^2}=\sqrt{1} \\ =1 \end{gathered}$$

Which gives

$$\begin{gathered} -i=1(0-i) \\ =(\cos 270°-i\sin 270°) \end{gathered}$$

Step 2. Using De Moivre's Theorem.

We get

$$\begin{gathered} z_k=\sqrt[5]{1}\left[\cos \right(\frac{270}{5}+\frac{360k}{5})+i\sin (\frac{270}{5}+\frac{360k}{5}\left)\right] \\ =\left[\cos \right(54°+72°k)+i\sin (54°+72°k\left)\right];k=0,1,2,3,4 \end{gathered}$$

Step 3. Substitute k=0,1,2,3,4 in zk=cos(54+72k)+isin(54+72k).

We get

$$\begin{gathered} z_0=\cos 54+i\sin 54 \\ {z}_1=\cos 126+i\sin 126 \\ {z}_2=\cos 198+i\sin 198 \\ {z}_3=\cos 270+i\sin 270 \\ {z}_4=\cos 342+i\sin 342 \end{gathered}$$

So we have found the complex fifth roots of$-i$.