Q26P - Mathematical Methods in Physical Sciences

Question

Find the values of the indicated roots.

 $\sqrt[\mathbf{5}]{\mathbf{i}}$

Step-by-Step Solution

Answer

The values of the complex number $\sqrt[5]{i}$are; 

$$\begin{gathered} z_0=0.95+0.31i \\ {z}_1=i \\ {z}_2=-0.95+0.31i \\ {z}_3=-0.588-0.81i \\ {z}_4=0.588-0.81i \end{gathered}$$ 

Step 1: Given Information

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

Step 2: Definition of the 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 = i 

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

 

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

Where k = 0,1,2,3,4 

And n = 5  

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

 

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

 

Solve z and $\theta$ for $k=0,1.$

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

Solve z and $\theta$ for k = 2,3 .

$$\begin{gathered} {\theta }_2=\frac{9\mathrm{\pi }}{10} \\ {z}_2=e^{(9\mathrm{\pi i}/10)} \\ {\mathrm{\theta }}_3=\frac{13\mathrm{\pi }}{10} \\ {\mathrm{z}}_3={\mathrm{e}}^{13\mathrm{\pi i}/10} \end{gathered}$$ 

 

Solve z and $\theta$for k = 4.

$$\begin{gathered} {\vartheta }_4=\frac{17\mathrm{\pi }}{10} \\ {z}_4=e^{17\mathrm{\pi i}/10} \end{gathered}$$

Step 4: Solving the Cartesian form of root

Solve for $z_0$ 

$$\begin{gathered} z_0=\left[\cos \left(\frac{\mathrm{\pi }}{10}\right)+i \sin \left(\frac{\mathrm{\pi }}{10}\right)\right] \\ =0.95+0.31i \end{gathered}$$  

 

Solve for $z_1$.

$$\begin{gathered} z_1=\left[\cos \left(\frac{\mathrm{\pi }}{10}\right)+i \sin \left(\frac{\mathrm{\pi }}{10}\right)\right] \\ =i \end{gathered}$$ 

 

Solve for $z_2$.

$$\begin{gathered} z_2=\left[\cos \left(\frac{9\mathrm{\pi }}{10}\right)+i \sin \left(\frac{9\mathrm{\pi }}{10}\right)\right] \\ =-0.95+0.31i \end{gathered}$$ $$\begin{gathered} z_2=\left[\cos \left(\frac{9\mathrm{\pi }}{10}\right)+i \sin \left(\frac{9\mathrm{\pi }}{10}\right)\right] \\ =0.95+0.31i \end{gathered}$$

Solve for $z_3$.

$$\begin{gathered} z_3=\left[\cos \left(\frac{13\mathrm{\pi }}{10}\right)+i \sin \left(\frac{13\mathrm{\pi }}{10}\right)\right] \\ =-0.588-0.81i \end{gathered}$$

Solve for $z_4$.

 $$\begin{gathered} z_4=\left[\cos \left(\frac{17\mathrm{\pi }}{10}\right)+i \sin \left(\frac{17\mathrm{\pi }}{10}\right)\right] \\ =-0.588-0.81i \end{gathered}$$

 

Hence, the values of the complex number $\sqrt[5]{i}$are; 

$$\begin{gathered} z_0=0.95+0.31i \\ {z}_1=i \\ {z}_2=-0.95+0.31i \\ {z}_3=-0.588-0.81i \\ {z}_4=0.588-0.81i \end{gathered}$$