Prove that `z=i^(i)` is purely real

Suppose z = x + iy and w = (1-iz)/(z-i) Find absw . If absw = 1, prove that z is purely real

Prove that (1 + i)^4 = -4

Prove that the complex number (3+2i)/(2-3i) + (3-2i)/(2+3i) is purely real

Find real theta such that (3+2 i sin theta)/(1-2 i sin theta) is purely real:

If (w- bar(w)z)/(1-z) is purely real where w= alpha + ibeta, beta ne 0 and z ne 1 , then the set of the values of z is

prove that (-1 + i)^2 = -2i

Prove that abs((1+i)/(1-i)) = 1

If z=|(-5,3+4i,5-7i),(3-4i,6,8+7i),(5+7i,8-7i,9)| then z is a)Purely real b)Purely imaginary c)a + ib where a!=0, b != 0 d)a + ib, where b = 4

Prove that abs((a+i)^2/(2a-i)) = (a^2+1)/(sqrt(4a^2+1))

Prove that (1+i)^4(1+1/i)^4=16