Question

w, w^2 , 1 are the roots of the 1
w^w = .....

Answer 1

if w, w^2 , 1 are all of the roots, then w is the cubed root of 1
if we write 1 in exponential form
\[ e^{2 \pi i} = 1\]
then w is
\[ w = e^{\frac{2 \pi}{3} i }\]
in rectangular a+bi form this is
\[ w = -\frac{1}{2} + \frac{\sqrt{3}}{2} i \]

thus we can write
\[ w^w= \left( e^{\frac{2 \pi}{3} i }\right)^{ -\frac{1}{2} + \frac{\sqrt{3}}{2} i } \]
use
\[ \left(a^b\right)^c = a^{bc} \]
so simplify

Answer 2

w^w = e^(-pi/3 * i - pi/sqrt(3))
How will I get rid of this i ? the question is asking for a value.

Answer 3

the answer is complex
you could write it in the form
\[ A \exp ( i \theta) \]
or
\[ A\cos(\theta) + i A\sin(\theta) \]
here A is \( \exp( -\pi/\sqrt{3}) \)
and theta = -pi/3

Answer 4

Thanks a lot.