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This image shows a mathematical expression involving the fifth root of -1:

\[
\sqrt[5]{-1} = \sqrt[5]{(-1)^5} = (-1)^{\frac{5}{5}} = (-1)^1 = -1
\]

Here's a breakdown of how to solve it:

1. **Expression**: The fifth root of \(-1\) is written as \(\sqrt[5]{-1}\). You can express \(-1\) as \((-1)^1\).

2. **Simplification under the root**: The next step rewrites \(\sqrt[5]{-1}\) as \(\sqrt[5]{(-1)^5}\), since raising a number to the power of 5 and then taking the fifth root "cancels" each other out.

3. **Simplify the powers**: The term \(\sqrt[5]{(-1)^5}\) simplifies to \((-1)^{\frac{5}{5}}\), which is equal to \((-1)^1\).

4. **Final result**: Since \((-1)^1 = -1\), the final answer is \(-1\).

### Conclusion:
\[
\sqrt[5]{-1} = -1
\]

Would you like any further details or clarifications on this?

Here are some related questions to deepen your understanding:
1. How does the property of roots and exponents apply to negative numbers?
2. What happens when you take the even root of negative numbers?
3. Can \((-1)\) raised to an even power ever give a negative result?
4. How would this process change for a cube root of \(-1\)?
5. Can you express other roots like \(\sqrt[3]{-1}\) similarly?

**Tip**: When working with roots and exponents, always check whether the root is even or odd. Odd roots of negative numbers give negative results, while even roots do not give real results (in the real number system).

Learn how to solve the fifth root of -1 with a step-by-step breakdown. This problem covers algebraic concepts including exponentiation, roots, and properties of negative numbers. Suitable for high school students in Grades 10-12.

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