Question

Which of the following is equivalent to the complex number i^35

Answer 1

-i

Explanation:

The imaginary unit i is defined by the property that
`i^2=-1`. With that in mind, we can list the first few powers of i:

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

Notice how
`i^0` is the same as
`i^4`? The powers of i are cyclic: they repeat every 4 powers. In general, for any power of i
`i^n`:

`i^n=1` if
`n/4` has remainder 0

`i^n=i` if
`n / 4` has remainder 1

`i^n=-1` if
`n/4` has remainder 2

`i^n=-i` if
`n/4` has remainder 3

`35/4=8R3`, so
`i^(35)=-i`