To evaluate \((-1)^{1/5}\), you are looking for the fifth root of \(-1\).

The fifth root of \(-1\) is a complex number. Specifically, in the context of complex numbers, it can be expressed using Euler's formula:

\[
(-1)^{1/5} = e^{i\pi/5}
\]

Calculating this gives:

\[
(-1)^{1/5} = \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)
\]

This results in one of the five fifth roots of \(-1\). The other roots can be obtained by adding multiples of \(2\pi/5\):

1. \((-1)^{1/5}\)
2. \((-1)^{1/5} e^{2\pi i/5}\)
3. \((-1)^{1/5} e^{4\pi i/5}\)
4. \((-1)^{1/5} e^{6\pi i/5}\)
5. \((-1)^{1/5} e^{8\pi i/5}\)

In real terms, the principal root is:

\[
\text{Principal root} = \cos\left(\frac{\pi}{5}\right) + i \sin\left(\frac{\pi}{5}\right)
\]

Would you like more details on any part of this? Here are some related questions:

1. How do you calculate the fifth root of other negative numbers?
2. What are the other fourth or fifth roots of real numbers?
3. How does Euler's formula relate to roots of negative numbers?
4. Can you explain how to graph complex numbers?
5. What is the significance of roots in polynomial equations?

**Tip:** When dealing with complex roots, it's often helpful to visualize them on the complex plane.

Discover how to evaluate (-1)^{1/5} using complex numbers and Euler's formula. This detailed guide explores the fifth roots of -1 and provides insight into their graphical representation on the complex plane. Ideal for advanced students in Grades 11-12.

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