To find the fifth root of a complex number $Z = -5 + 7i$, we follow these steps:

1. Convert the complex number into polar form:
   A complex number $Z = x + yi$ can be expressed in polar form as: $Z = r(\cos \theta + i \sin \theta)$ where: $r = \sqrt{x^2 + y^2}$ and the argument $\theta$ is given by: $\theta = \tan^{-1} \left( \frac{y}{x} \right)$

   For $Z = -5 + 7i$:

   • $x = -5$
   • $y = 7$

   We calculate $r$ and $\theta$.

2. Find the fifth roots:
   Once we have the polar form $Z = r(\cos \theta + i \sin \theta)$, the fifth roots of $Z$ are given by: $Z_k = \sqrt[5]{r} \left( \cos \frac{\theta + 2k\pi}{5} + i \sin \frac{\theta + 2k\pi}{5} \right)$ for $k = 0, 1, 2, 3, 4$.

Let's calculate it step by step.

Step 1: Convert to polar form

Calculate the modulus $r$:

$r = \sqrt{(-5)^2 + 7^2} = \sqrt{25 + 49} = \sqrt{74}$

Calculate the argument $\theta$:

$\theta = \tan^{-1} \left( \frac{7}{-5} \right)$ This gives the principal angle, but since the complex number is in the second quadrant (because $x = -5$ and $y = 7$), we adjust $\theta$ accordingly.

Now I will compute these values.We have the following values for the complex number $Z = -5 + 7i$:

• Modulus $r = 8.60$
• Argument $\theta = 2.19$ radians.

Step 2: Find the fifth roots

The general formula for the $k$-th fifth root of $Z$ is: $Z_k = \sqrt[5]{r} \left( \cos \frac{\theta + 2k\pi}{5} + i \sin \frac{\theta + 2k\pi}{5} \right)$ where $k = 0, 1, 2, 3, 4$.

Now, I will compute the fifth roots.The five fifth roots of $Z = -5 + 7i$ are:

$Z_0 = 1.39 + 0.65i$ $Z_1 = -0.19 + 1.53i$ $Z_2 = -1.51 + 0.29i$ $Z_3 = -0.74 - 1.35i$ $Z_4 = 1.05 - 1.12i$

These are the five distinct fifth roots of the complex number.

Would you like more details, or do you have any other questions? Here are some related topics:

1. What is De Moivre’s Theorem and how is it applied to find roots?
2. How do you convert between rectangular and polar forms of complex numbers?
3. How can you visualize complex roots on an Argand diagram?
4. Why do complex numbers have multiple roots?
5. How do complex numbers behave under exponentiation?

Tip: When finding roots of complex numbers, always remember that the number of distinct roots corresponds to the order of the root (e.g., 5 roots for the fifth root).