$$ \sqrt[3]{\imaginaryI}^{\placeholder{}} $$$$  $$

The expression you provided is $ \sqrt[3]{\imaginaryI} $. This represents the cube root of the imaginary unit $ \imaginaryI $.

### Step-by-Step Solution:

1. **Expressing $ \imaginaryI $ in Polar Form:**
   - The imaginary unit $ \imaginaryI $ can be written in polar form as $ \imaginaryI = e^{\imaginaryI \frac{\pi}{2}} $.

2. **Taking the Cube Root:**
   - To find $ \sqrt[3]{\imaginaryI} $, we take the cube root of the polar form:
     \[
     \sqrt[3]{\imaginaryI} = \sqrt[3]{e^{\imaginaryI \frac{\pi}{2}}}
     \]
   - This can be rewritten as:
     \[
     \sqrt[3]{\imaginaryI} = e^{\imaginaryI \frac{\pi}{6}}
     \]

3. **Finding All Cube Roots:**
   - The general solution for cube roots in complex numbers involves adding multiples of $ \frac{2\pi k}{3} $ (where $ k = 0, 1, 2 $) to the exponent:
     \[
     \sqrt[3]{\imaginaryI} = e^{\imaginaryI \left(\frac{\pi}{6} + \frac{2\pi k}{3}\right)} \quad \text{for } k = 0, 1, 2
     \]
   - These correspond to the three cube roots of $ \imaginaryI $.

### Cube Roots of $ \imaginaryI $:
The three distinct cube roots of $ \imaginaryI $ are:
\[
e^{\imaginaryI \frac{\pi}{6}}, \quad e^{\imaginaryI \frac{5\pi}{6}}, \quad e^{\imaginaryI \frac{3\pi}{2}}
\]

These can also be expressed in rectangular form:
1. $ \frac{\sqrt{3}}{2} + \frac{1}{2}\imaginaryI $
2. $ -\frac{\sqrt{3}}{2} + \frac{1}{2}\imaginaryI $
3. $ -\imaginaryI $

### Do you want any further details or have any questions?

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Here are 5 related questions:

1. What is the general formula for finding the $n$th roots of a complex number?
2. How do you represent complex numbers in polar form?
3. What are the properties of the imaginary unit $ \imaginaryI $?
4. How do you convert between rectangular and polar forms of complex numbers?
5. What is the significance of the argument (angle) in the polar form of a complex number?

**Tip:** Remember that for any complex number $ z $, the $ n $th roots are equally spaced around the complex plane, forming a regular $ n $-sided polygon.

Explore the cube roots of the imaginary unit $ \imaginaryI $ with a detailed step-by-step solution. Learn about complex numbers, cube roots, and their representations in polar and rectangular forms. Suitable for advanced high school students and anyone interested in complex number theory.

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