91Ó°ÊÓ

The given complex number is$\sqrt[5]{-4-4i}$ .

The numbers that are presented in the form of x + iy where, ' x ' is real numbers and

' ib ' is an imaginary number, those numbers are referred to as called Complex numbers.

Let,

$$\begin{gathered} w=\sqrt[5]{-4-4i} \\ z=-4-4i \\ z=r{e}^{\left(\mathrm{θ}i\right)} \\ =4\sqrt{2}{e}^{\frac{5\mathrm{π}}{4}} \end{gathered}$$

Write the root of in different form.

$w_k=r^{\frac{1}{n}}{e}^{{\mathrm{θ}}_k}$

Where${\mathrm{θ}}_k$ can be represented as

$$\begin{gathered} {\mathrm{θ}}_k=\frac{{\displaystyle \frac{5\mathrm{π}}{4}}+2k\mathrm{π}}{5} \\ k=0,1,2,3,4 \end{gathered}$$

$$\begin{gathered} {\mathrm{θ}}_o=\frac{\mathrm{Ï€}}{4} \\ ⇒w_0=\sqrt{2}{e}^{\frac{\mathrm{Ï€¾\pm }}{4}} \\ {\mathrm{θ}}_1=\frac{13\mathrm{Ï€}}{20} \\ ⇒w_1=\sqrt{2}{e}^{\frac{13\mathrm{Ï€¾\pm }}{20}} \\ {\mathrm{θ}}_2=\frac{21\mathrm{Ï€}}{20} \\ ⇒w_2=\sqrt{2}{e}^{\frac{21\mathrm{Ï€¾\pm }}{20}} \\ {\mathrm{θ}}_3=\frac{29\mathrm{Ï€}}{20} \\ ⇒w_3=\sqrt{2}{e}^{\frac{29\mathrm{Ï€¾\pm }}{20}} \\ {\mathrm{θ}}_4=\frac{37\mathrm{Ï€}}{20} \\ ⇒w_2=\sqrt{2}{e}^{\frac{37\mathrm{Ï€¾\pm }}{20}} \end{gathered}$$

$$\begin{gathered} w_0=\sqrt{2}\cos \left(\frac{\mathrm{Ï€}}{4}\right)+i\sqrt{2}\sin \left(\frac{\mathrm{Ï€}}{4}\right) \\ =1+i \\ {w}_1=\sqrt{2}\cos \left(\frac{13\mathrm{Ï€}}{20}\right)+i\sqrt{2}\sin \left(\frac{13\mathrm{Ï€}}{20}\right) \\ =-0.64+1.26i \\ {w}_2=\sqrt{2}\cos \left(\frac{21\mathrm{Ï€}}{20}\right)+i\sqrt{2}\sin \left(\frac{21\mathrm{Ï€}}{20}\right) \\ =-1.4-0.22i \\ {w}_3=\sqrt{2}\cos \left(\frac{29\mathrm{Ï€}}{20}\right)+i\sqrt{2}\sin \left(\frac{29\mathrm{Ï€}}{20}\right) \\ =-0.22-1.40i \\ {w}_4=\sqrt{2}\cos \left(\frac{37\mathrm{Ï€}}{20}\right)+i\sqrt{2}\sin \left(\frac{37\mathrm{Ï€}}{20}\right) \\ =1.26-0.64i \end{gathered}$$

Hence, the value of $\sqrt[5]{-4-4i}$is

$$\begin{gathered} w_0=1+i \\ {w}_1=-0.64+1.26i \\ {w}_2=-1.4-0.22i \\ {w}_3=-0.22-1.40i \\ {w}_4=1.26-0.64i \end{gathered}$$