The equation you provided is:

\[
z = i^i + z \cdot i
\]

Let's break it down and solve it step by step.

1. **First term:** \( i^i \)
   - The imaginary unit \( i \) is defined as \( \sqrt{-1} \). When raised to itself, \( i^i \) is a complex number.
   - Using the Euler's formula, \( i = e^{i\frac{\pi}{2}} \), we can compute \( i^i \).

   \[
   i^i = e^{-\frac{\pi}{2}} \approx 0.20788
   \]

   So, \( i^i \) is a real number.

2. **Second term:** \( z \cdot i \)
   - This represents the product of the complex variable \( z \) and the imaginary unit \( i \). If \( z = a + bi \), multiplying by \( i \) gives:

   \[
   z \cdot i = (a + bi) \cdot i = ai + b(-1) = -b + ai
   \]

Thus, we now have:

\[
z = 0.20788 + (-b + ai)
\]

This gives the complex number form of \( z \). To find specific values for \( a \) and \( b \), you'd need additional conditions or context for the value of \( z \).

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

Here are 5 related questions to explore:
1. What is Euler's formula and how is it applied to compute \( i^i \

Learn how to solve the complex equation z = i^i + z * i using advanced concepts in complex numbers and Euler's formula. This step-by-step solution breaks down the calculation of i raised to the power of i, and explains the manipulation of complex variables. Suitable for advanced math students in Grades 11-12.

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