Question 1165887: Express the roots of (-14+3i)^-2/5 complex number in polar form.
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! ```python?code_reference&code_event_index=2
import numpy as np
# W = -14 + 3i
# Modulus r = sqrt(14^2 + 3^2) = sqrt(205)
r_squared = 205
R = 1 / r_squared # Modulus of C = W^-2
# Argument calculation for W
alpha = np.arctan(3/14) # Reference angle
# phi_W = pi - alpha (Argument of W)
# Argument calculation for C = W^-2
# phi = -2 * phi_W = -2 * (pi - alpha) = -2*pi + 2*alpha. We use the principal argument 2*alpha.
phi = 2 * alpha
# Modulus of the roots (R_root)
R_root = R**(1/5)
R_root_decimal = R_root # (1/205)^(1/5)
# Calculate phi in degrees for understanding
phi_degrees = np.degrees(phi)
# Calculate the argument for k=0
theta_0 = phi / 5
# Calculate the arguments for k=0, 1, 2, 3, 4 (in radians)
k_values = np.arange(5)
theta_k = (phi + 2 * np.pi * k_values) / 5
print(f"Modulus of the roots (R_root): {R_root_decimal}")
print(f"Base angle phi (radians): {phi}")
print(f"Argument theta_k (radians): {theta_k}")
```
```text?code_stdout&code_event_index=2
Modulus of the roots (R_root): 0.34486508519906256
Base angle phi (radians): 0.4221866664454931
Argument theta_k (radians): [0.08443733 1.34107439 2.59771146 3.85434852 5.11098558]
```
The problem asks for the roots of the complex number $Z = (-14+3i)^{-2/5}$ in polar form. We interpret this as finding the **five fifth roots** of the complex number $C = (-14+3i)^{-2}$.
The polar form of a complex number is $Z = R e^{i\theta}$. The $n$-th roots of $Z$ are given by De Moivre's Theorem:
$$Z_k = \sqrt[n]{R} \cdot e^{i \left(\frac{\theta + 2\pi k}{n}\right)}, \quad \text{for } k=0, 1, 2, \dots, n-1$$
Here, $n=5$.
## 1\. Convert the Base Complex Number to Polar Form
Let $W = -14 + 3i$. We first convert $W$ to polar form $W = r e^{i\theta}$.
$$r = \sqrt{(-14)^2 + 3^2} = \sqrt{196 + 9} = \sqrt{205}$$
$$\theta = \pi - \tan^{-1}\left(\frac{3}{14}\right) \quad (\text{Since } W \text{ is in QII})$$
Now we find $C = W^{-2}$:
* **Modulus ($R$):**
$$R = r^{-2} = (\sqrt{205})^{-2} = \frac{1}{205}$$
* **Argument ($\phi$):**
$$\phi = -2\theta = -2\left(\pi - \tan^{-1}\left(\frac{3}{14}\right)\right)$$
Using the periodicity of the argument, we take the principal argument, $\phi = 2 \tan^{-1}\left(\frac{3}{14}\right)$.
$$\phi \approx 2 \times 0.211093 \approx 0.42219 \text{ radians}$$
Thus, $C \approx \frac{1}{205} e^{i(0.42219)}$.
## 2\. Find the Five Fifth Roots ($Z_k$)
The roots $Z_k$ are the five distinct values of $C^{1/5}$.
### A. Modulus of the Roots ($R_{\text{root}}$)
$$R_{\text{root}} = \sqrt[5]{R} = \sqrt[5]{\frac{1}{205}} \approx \mathbf{0.3449}$$
### B. Arguments of the Roots ($\theta_k$)
The arguments are calculated using the formula:
$$\theta_k = \frac{\phi + 2\pi k}{5}, \quad \text{for } k=0, 1, 2, 3, 4$$
We use the exact value $\phi = 2 \tan^{-1}\left(\frac{3}{14}\right)$ and the approximation $\phi \approx 0.42219$.
| k | Argument $\theta_k$ (Exact Form) | Argument $\theta_k$ (Approx. Radians) |
| :---: | :---: | :---: |
| 0 | $\frac{1}{5} \left( 2 \tan^{-1}\left(\frac{3}{14}\right) \right)$ | $\mathbf{0.0844}$ |
| 1 | $\frac{1}{5} \left( 2 \tan^{-1}\left(\frac{3}{14}\right) + 2\pi \right)$ | $\mathbf{1.3411}$ |
| 2 | $\frac{1}{5} \left( 2 \tan^{-1}\left(\frac{3}{14}\right) + 4\pi \right)$ | $\mathbf{2.5977}$ |
| 3 | $\frac{1}{5} \left( 2 \tan^{-1}\left(\frac{3}{14}\right) + 6\pi \right)$ | $\mathbf{3.8543}$ |
| 4 | $\frac{1}{5} \left( 2 \tan^{-1}\left(\frac{3}{14}\right) + 8\pi \right)$ | $\mathbf{5.1110}$ |
The roots of the complex number $(-14+3i)^{-2/5}$ in polar form are:
$$Z_k = \frac{1}{\sqrt[5]{205}} \cdot e^{i \left( \frac{2 \tan^{-1}(3/14) + 2\pi k}{5} \right)}, \quad \text{for } k=0, 1, 2, 3, 4$$
|
|
|
