What is $$\sqrt {{{1 + {\omega ^2}} \over {1 + \omega }}} $$ equal to, where $$\omega$$ is the cube root of unity?

What is $${\omega ^{100}} + {\omega ^{200}} + {\omega ^{300}}$$ equal to, where $$\omega$$ is the cube root of unity?

If $${\mathop{\rm Re}\nolimits} \left( {{{z - 1} \over {z + 1}}} \right) = 0$$, where z = x + iy is a complex number, then which one of the following is correct?

If $$z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^{107}} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^{107}}$$, then what is the imaginary part of z equal to?