Powers of imaginary partPowers of imaginary part.

Expert Answers
justaguide eNotes educator| Certified Educator

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.

For example:

i^3 = -1*i = -i

i^4 = (-1)^2 = 1

giorgiana1976 | Student

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:

 i^1997= i^1996*i=(i^4)^499*i=1*i=i