Q3MP - Mathematical Methods in Physical Sciences

Question

Find one or more values of each of the following complex expressions and compare with a computer solution.

$\sqrt[\mathbf{5}]{\mathbf{-}\mathbf{4}\mathbf{-}\mathbf{4}\mathbf{i}}$

Step-by-Step Solution

Answer

The value of  $\sqrt[5]{-4-4i} is$

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

Step 1: Given Information.

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

Step 2: Definition of Complex Number.

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.

Step 3: Find the value of - 4 - 4 i 5 .

Let,

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

 

Write the root of in different form. 

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

Where ${\theta }_k$ can be represented as 

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

$$\begin{gathered} {\theta }_o=\frac{\mathrm{\pi }}{4} \\ \Rightarrow w_0=\sqrt{2}{e}^{\frac{\mathrm{\pi i}}{4}} \\ {\theta }_1=\frac{13\mathrm{\pi }}{20} \\ \Rightarrow w_1=\sqrt{2}{e}^{\frac{13\mathrm{\pi i}}{20}} \\ {\theta }_2=\frac{21\mathrm{\pi }}{20} \\ \Rightarrow w_2=\sqrt{2}{e}^{\frac{21\mathrm{\pi i}}{20}} \\ {\theta }_3=\frac{29\mathrm{\pi }}{20} \\ \Rightarrow w_3=\sqrt{2}{e}^{\frac{29\mathrm{\pi i}}{20}} \\ {\theta }_4=\frac{37\mathrm{\pi }}{20} \\ \Rightarrow w_2=\sqrt{2}{e}^{\frac{37\mathrm{\pi i}}{20}} \end{gathered}$$ 

Step 4: Write the in the rectangular form of numbers.

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

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

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