Simplify. \( (-1)^{-\frac{1}{5}} \)

Did you know that any real number raised to a negative exponent can be expressed as the reciprocal of the number raised to the positive exponent? So, you can rewrite \( (-1)^{-\frac{1}{5}} \) as \( \frac{1}{(-1)^{\frac{1}{5}}} \). Now, finding the fifth root of \(-1\) gives us a lovely twist—it's actually \(-1\) because \(-1 \times -1 \times -1 \times -1 \times -1 = -1\)! Additionally, the properties of exponents allow us to explore complex numbers here as well. When dealing with roots and negative bases, it opens up the world of complex numbers. If we venture beyond real numbers, we can express \( (-1)^{\frac{1}{5}} \) using Euler's formula or De Moivre's theorem, taking us through fascinating realms of \( i \), resulting in multiple complex solutions too! It's math with a twist!