Q25P - Mathematical Methods in Physical Sciences

Question

Find the values of the indicated roots:

$\sqrt[\mathbf{5}]{\mathbf{-}\mathbf{1}\mathbf{-}\mathbf{i}}$

Step-by-Step Solution

Answer

The values of the complex number $\sqrt[5]{-1-\mathrm{i}}$ are:

$$\begin{gathered} z_0=0.758+0.758i \\ {z}_1=-0.487+0.955i \\ {z}_2=-1.059-0.167i \\ {z}_3=-0.168-1.059i, \\ {z}_4=0.487-0.955i \end{gathered}$$

Step 1: Given Information

The given expression is $\sqrt[5]{-1-\mathrm{i}}$.

Step 2: Definition of Complex Number

Complex numbers are represented in terms of real numbers and imaginary numbers; a complex  can be written in the form of: 

$\mathit{z}\mathbf{=}\mathit{a}\mathbf{+}\mathit{i}\mathit{b}$  

 

Here a and b are real numbers, and i is the imaginary number which is known as iota, whose value is $\sqrt{\mathbf{-}\mathbf{1}}$.

Step 3: Solving the Equation

Let $z=-1-i$ .

The exponential form of z is given by $z=r\times e^{\left({\theta }_i\right)}$ .                                                                                

 

Find the modulus of the complex number z.

 

$$\begin{gathered} r=\sqrt{1+1} \\ =\sqrt{2} \end{gathered}$$ 

 

Find the angle of the complex number z .

 

$$\begin{gathered} \theta =arc\tan \left(1\right) \\ =\frac{\mathrm{\pi }}{4} \end{gathered}$$ 

 

Find the angle in the 3^rd quadratic.

$$\begin{gathered} \theta =\mathrm{\pi }+\frac{\mathrm{\pi }}{4} \\ =\frac{5\mathrm{\pi }}{4} \end{gathered}$$

Hence the root is $z_k=r^{\frac{1}{n}}exp\left({\theta }_{k}i\right)$ .

 

Where $k=0,1,2,3,4 \mathrm{And} n=5$.

 

Angle ${\theta }_k$ is written as ${\theta }_k=\frac{{\displaystyle \frac{5\mathrm{\pi }}{4}}+2\mathrm{\pi k}}{5}$ .

Step 4- Roots in Exponential Form

Find the roots of the complex number z for different values of $\theta$ .

 

Solve z and $\theta$ for $k=0,1$.

$$\begin{gathered} {\theta }_0=\frac{\mathrm{\pi }}{4} \\ {z}_0=2^{\frac{1}{10}}{e}^{\left(\mathrm{\pi }/4\right)} \\ {\theta }_1=\frac{13\mathrm{\pi }}{20} \\ {z}_1=2^{\frac{1}{10}}{e}^{\left(13\mathrm{\pi }/20\right)} \end{gathered}$$ 

 

Solve z and $\theta$ for $k=2,3$.

$$\begin{gathered} {\theta }_0=\frac{21\mathrm{\pi }}{20} \\ {z}_2=2^{\frac{1}{10}}{e}^{\left(21\mathrm{\pi }/20\right)} \\ {\theta }_3=\frac{29\mathrm{\pi }}{20} \\ {z}_3=2^{\frac{1}{10}}{e}^{29\mathrm{\pi }/20} \end{gathered}$$ 

 

Solve z and $\theta$ for $k=4$.

$$\begin{gathered} {\theta }_4=\frac{73\mathrm{\pi }}{20} \\ {z}_4=2^{\frac{1}{10}}{e}^{73\mathrm{\pi }/20} \end{gathered}$$

Step 5: Solving the Cartesian form of root

Solve for $z_0$

$$\begin{gathered} z_0=1.072\left[\cos \left(\frac{\mathrm{\pi }}{4}\right)+i \sin \left(\frac{\mathrm{\pi }}{4}\right)\right] \\ =0.758+0.758i \end{gathered}$$ 

 

Solve for $z_1$ .

$$\begin{gathered} z_1=1.072\left[\cos \left(\frac{13\mathrm{\pi }}{20}\right)+i \sin \left(\frac{13\mathrm{\pi }}{20}\right)\right] \\ =0.487+0.955i \end{gathered}$$ 

 

Solve for $z_2$ .

$$\begin{gathered} z_2=1.072\left[\cos \left(\frac{21\mathrm{\pi }}{20}\right)+i \sin \left(\frac{21\mathrm{\pi }}{20}\right)\right] \\ =1.059-0.167i \end{gathered}$$ 

 

Solve for $z_3$ .

$$\begin{gathered} z_3=1.072\left[\cos \left(\frac{29\mathrm{\pi }}{4}\right)+i \sin \left(\frac{\mathrm{\pi }}{4}\right)\right] \\ =0.168+1.059i \end{gathered}$$ 

 

Solve for $z_4$ .

$$\begin{gathered} z_4=1.072\left[\cos \left(\frac{73\mathrm{\pi }}{20}\right)+i \sin \left(\frac{73\mathrm{\pi }}{20}\right)\right] \\ =0.487-0.955i \end{gathered}$$

Hence the values of the complex number $\sqrt[5]{-1-i}$  are:

$$\begin{gathered} z_0=0.758+0.758i \\ {z}_1=-0.487+0.955i \\ {z}_2=-1.059-0.167i \\ {z}_3=-0.168-1.059i, \\ {z}_4=0.487-0.955i \end{gathered}$$