Question

How do you simplify `i^21 + i^30`?

Answer 1

`i-1`

Explanation:

Observe that the powers of `i` are cyclic:

`i^0= 1`
`i^1 = i`
`i^2 = -1`
`i^3=-i`
`i^4 = 1`
`...`

So, to find high powers of `i`, we can calculate the modulus `4` Since `21 = 5*4 +1`, and `30 = 7*4 +2`, we have that `i^21=i^1=i`, and `i^30 = i^2 = -1`. So,

`i^21+i^30 = i-1`.