METHOD 1 (long)
z=2-4i=2(1-2i); z10=210(1-2i)10.
Coefficients in expansion:
1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1.
(1-2i)10=
1-10(2i)+45(2i)2-120(2i)3+210(2i)4-252(2i)5+210(2i)6-120(2i)7+45(2i)8-10(2i)9+(2i)10,
(1-2i)10=
1+45(2i)2+210(2i)4+210(2i)6+45(2i)8+(2i)10 -
(10(2i)+120(2i)3+252(2i)5+120(2i)7+10(2i)9),
(1-2i)10=1-45(22)+210(24)-210(26)+45(28)-210 -
i(10(2)-120(23)+252(25)-120(27)+10(29)),
(1-2i)10=1+45(28-22)-210(26-24)-210 -
i(10(2+29)-120(23+27)+252(25)),
(1-2i)10=1+45(252)-210(48)-1024 -
i(10(514)-120(136)+252(32)),
(1-2i)10=1+11340-10080-1024-i(5140-16320+8064),
(1-2i)10=237+3116i.
z10=1024(237+3116i)=242688+3190784i.
METHOD 2 (short)
z=2-4i=reiθ=r(cosθ+isinθ); z10=r10e10iθ=r10(cos(10θ)+isin(10θ)).
rcosθ=2, rsinθ=-4; r2=20, tanθ=-2, θ=-1.10715 radians approx, 10θ=-11.0715 radians.
z10=205(0.07584+0.99712i)=242688+3190784i.
The two methods concur in their results (using a calculator for accuracy in Method 2).