Fun with Cube root of unity!

$\begin{aligned} x & = \Re \left( \omega^\omega \right) \\ & = \Re \left(\exp \left(\frac {2\pi}3 i\cdot e^{\frac {2\pi}3 i}\right) \right) \\ & = \Re \left(\exp \left(\frac {2\pi}3 i \left(\cos \frac {2 \pi}3 + i \sin \frac {2 \pi}3\right)\right) \right) \\ & = \Re \left(\exp \left(\frac {2\pi}3 i \left(-\frac 12 + \frac {\sqrt 3}2 i \right)\right) \right) \\ & = \Re \left(\exp \left(-\frac {\pi}3 i - \frac \pi{\sqrt 3} \right) \right) \\ & = \Re \left(e^{-\frac \pi{\sqrt 3}} \left(\cos \frac {\pi}3 - i \sin \frac {\pi}3 \right) \right) \\ & = e^{-\frac \pi{\sqrt 3}} \cos \frac {\pi}3 \\ & = \frac 12 e^{-\frac \pi{\sqrt 3}} \\ & \approx \boxed{0.0815} \end{aligned}$

Ha ha nice one.

Firstly w = e^(i2pi/3)

then w in power is (-1+iroot3)/2

Then used euler's form of complex numbers

to get the complex number as (1-iroot3)/2[(e)^(pi/root3)]

calculating this is a pretty herculian task :P