To find $\sqrt[n]{\imaginaryI}$, we are looking for the $n$-th roots of the imaginary unit $\imaginaryI$.

The $n$-th roots of any complex number $z = re^{i\theta}$ (in polar form) can be expressed as:

[ \sqrt[n]{z} = \sqrt[n]{r} \cdot e^{i\left(\frac{\theta + 2k\pi}{n}\right)}, \quad k = 0, 1, \ldots, n-1 ]

For $\imaginaryI$, we can write it in polar form as:

$\imaginaryI = e^{i\frac{\pi}{2}}$

Thus, the $n$-th roots of $\imaginaryI$ are:

$\sqrt[n]{\imaginaryI} = e^{i\left(\frac{\frac{\pi}{2} + 2k\pi}{n}\right)}, \quad k = 0, 1, \ldots, n-1$

This simplifies to:

$\sqrt[n]{\imaginaryI} = e^{i\left(\frac{\pi}{2n} + \frac{2k\pi}{n}\right)}, \quad k = 0, 1, \ldots, n-1$

These roots are $n$ equally spaced points on the complex unit circle.

Would you like a specific example or further details on this? Here are some related questions:

1. What are the square roots ($n = 2$) of $\imaginaryI$?
2. How can you express complex numbers in polar form?
3. How do the $n$-th roots of unity relate to the $n$-th roots of $\imaginaryI$?
4. How do you find the $n$-th roots of a complex number not on the unit circle?
5. What is the significance of the argument (angle) in polar form?

Tip: The $n$-th roots of a complex number are evenly spaced around a circle in the complex plane, centered at the origin, with radius equal to the $n$-th root of the number's magnitude.