The <polar form> $1+i=\sqrt2e^{i\pi/4}$ gives $(1+i)^6=-8i$. Multiplication therefore scales the <modulus> by $8$ and rotates the <complex argument> through $-\pi/2$:
$$
\boxed{|(1+i)^6z|=8\sqrt{a^2+b^2},\qquad\arg((1+i)^6z)=\arg(a+ib)-\frac\pi2\pmod{2\pi}.}
$$
Equivalently the result is $8b-8ia$, whose <complex argument> is $\operatorname{atan2}(-a,b)$. If $a=b=0$, its <modulus> is zero and its <complex argument> is undefined.

Solution

Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 1 / ii / Solution

 Solution (source code)