Rewrite (2+2i)^6 as
[2^3/2*(cos pi/4 + i sin pi/4)]^6
= 2^9*[cos 6pi/4 * i sin 6pi/4]
= 512* i sinpi/2
= -512 i
I see several error in your derivation.
Where did you get the angle of 7π/4 ?
I get an angle of π/4
√8(cos π/4 + i sin π/4_
= √8(1/√2 + i (1/√2)
= 2 + 2i
so by De Moivre's Theorem
= √8^6(cos (6π/4) + i sin (6π/4))
= 512( + i(-1))
= - 512i
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