Fun with Cube root of unity!

Algebra Level 5

Find \(\Re (\omega^\omega )\), where \( \omega\) is the cube root of unity with positive imaginary part. Consider the priciple branch for the answer.

Notation: \(\Re (z)\) denotes the real part of complex number \(z\).

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This is a part of my set Aniket's Challenges in Mathematics
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