Question 1184507
The problem with representing {{{(1+3i)^(-4)}}} with {{{(2e^((pi/3)*i))^(-4)}}} is that {{{1+3i <> 2e^((pi/3)*i) = 1+ sqrt(3)*i}}}.


The best way to evaluate {{{(1+3i)^(-4)}}} is to evaluate it directly:


{{{(1+3i)^(-4) = 1/(1+3i)^4  = (1/(1+3i)^4)*((1-3i)^4/(1-3i)^4)  = (1-3i)^4/10^4 = (1-12i-54+108i+81)/"10,000"  =(7+24i)/"2,500"}}}.


BUT, in case you actually meant {{{(1+sqrt(3)i)^(-4)}}}, then


{{{(1+sqrt(3)i)^(-4) = (2e^((pi/3)*i))^(-4) = 2^(-4)*e^(-(4*pi/3)*i) = (1/2^4)*(cos(-4*pi/3) + i*sin(- 4*pi/3) )= (1/16)(-1/2 + (sqrt(3)/2)*i) = -1/32 + ((sqrt(3))/32)*i}}}