1 + i2 + 3i5 + 8i13 + 21i34
Hmm, this sequence of numbers looks familiar. Fibonacci, is that you?
1 + (-1) + 3i(i4) + 8i(i4)3 + 21i2(i4)8
When you have an exponent of i, check to see if it 4 or less. If so, simplify. If not, find as many powers of 4 as you can, because each i4 equals 1, which will multiply out without consequence.
0 + 3i + 8i + 21(-1)-21 + 11i
Remember to put the real number portion first and the imaginary part second.