Derive formula for finding nth roots of complex number. 
Find the roots of following numbers and plot them in complex plane. 
1
ଵ/ହ


The plots above show the 5th roots of 1 and \( \frac{1}{5} \) in the complex plane:

1. **5th Roots of 1**: The roots are evenly spaced points on the unit circle centered at the origin (with a radius of 1).
2. **5th Roots of \( \frac{1}{5} \)**: The roots are also evenly spaced points, but on a smaller circle centered at the origin, with a radius of \( \frac{1}{5^{1/5}} \).

These plots illustrate the symmetry and spacing of the roots around the origin.

Would you like more details on any step, or do you have any questions?

### Related Questions:
1. How do the arguments of the roots differ between \( z = 1 \) and \( z = \frac{1}{5} \)?
2. How does the magnitude of \( z \) affect the size of the circle on which the roots lie?
3. What would happen if we considered the 7th roots instead of the 5th roots?
4. How can we generalize this process to find the roots of any complex number?
5. How are the roots related to the concept of rotation in the complex plane?

#### Tip:
When finding the roots of a complex number, the geometric interpretation in the complex plane can provide valuable insights into the nature of the solutions.

Explore how to find the nth roots of complex numbers with a focus on 5th roots of 1 and 1/5. Understand the geometric interpretation in the complex plane and the symmetry of roots. Suitable for university-level mathematics students.

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