Write the expression in the standard form a + bi.
(1 - sqrt{5} i)^8
Let $\displaystyle z = 1 - \sqrt{5} \, i$.
You need to find $\displaystyle z^8$.
Note that $\displaystyle z^2 = (1 - \sqrt{5} \, i) (1 - \sqrt{5} \, i) = -4 - 2 \sqrt{5} \, i = -2 (2 + \sqrt{5} \, i)$.
Therefore $\displaystyle (z^2)^2 = z^4 = [ -2 (2 + \sqrt{5} \, i) ]^2 = 4 (2 + \sqrt{5} \, i)^2 = \, ....$
Therefore $\displaystyle (z^4)^2 = z^8 = \, .... $