All the powers of the imaginary part can be determined if we keep in mind the property that i is the square root of -1 or i^2 = -1.
Higher powers are easily resolved once this is done.
i^3 = -1*i = -i
i^4 = (-1)^2 = 1
The powers of the imaginary part of a complex number are:
i = i
i^2 = -1
i^3 = i^2*i = -1*i = -i
i^4 = 1
i^5 = i^4*i = 1*i = 1
We notice that after 4 steps, the powers are repeating.
This fact is important in calculating powers of imaginary part, that seems to be impossible to be calculated.
For instance, we'll have to calculate: i^1997
We'll try to write the exponent of i as a multiple of 4: