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The expression shown is \( (1)^{\frac{1}{8}} \), which represents the eighth root of 1. 

In the context of complex numbers, finding the eighth root of 1 means identifying all the complex solutions that satisfy the equation \( z^8 = 1 \). These solutions are known as the eighth roots of unity.

The general formula for finding the \(n\)-th roots of unity is:

\[
z_k = e^{i\frac{2\pi k}{n}}, \quad \text{for} \, k = 0, 1, 2, \ldots, n-1
\]

For the eighth roots of unity (\(n = 8\)), the solutions are:

\[
z_k = e^{i\frac{2\pi k}{8}} = e^{i\frac{\pi k}{4}}, \quad k = 0, 1, 2, \ldots, 7
\]

These correspond to the following values:

\[
z_0 = 1, \quad z_1 = e^{i\frac{\pi}{4}}, \quad z_2 = e^{i\frac{\pi}{2}}, \quad \ldots, \quad z_7 = e^{i\frac{7\pi}{4}}
\]

These points are evenly spaced on the unit circle in the complex plane, at angles of \( \frac{\pi}{4} \) radians apart.

Do you have any questions or want more details about this?

### Related Questions:
1. What are the geometric representations of the eighth roots of unity?
2. How do we represent the eighth roots of unity in rectangular form?
3. What is the general form for \(n\)-th roots of unity?
4. Can you explain how De Moivre's Theorem is used in finding roots of complex numbers?
5. How does Euler's formula relate to complex roots?

### Tip:
Always remember that the roots of unity are evenly spaced on the unit circle in the complex plane!

Explore how to find the eighth roots of 1 in the context of complex numbers. Learn the general formula for the nth roots of unity, the exponential form of complex numbers, and the geometric representation of roots on the unit circle.

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