Let z = (√3 + i)/(1 + i). Use the principal argument, in degrees, in (-180°, 180°]…

Polar forms. $\sqrt3 + i$ has modulus 2 and argument $30^\circ$; $1 + i$ has modulus $\sqrt2$ and argument $45^\circ$.

(a), (b) $|z| = \dfrac{2}{\sqrt2} = \sqrt{2} = 1.41$ and $\arg z = 30^\circ - 45^\circ = -15^\circ$.

(c) $z^n$ has argument $-15n^\circ$, real when $15n$ is a multiple of 180: the smallest is $n = 12$.

(d) $z^{12}$ has modulus $\left(\sqrt2\right)^{12} = 64$ and argument $-180^\circ$, so it is $-64$.

(e) Purely imaginary needs $15n = 90 + 180m$, that is $n = 6 + 12m$: $n = 6, 6 + 12, \dots, 90$, which is 8 values.