Question

How do you simplify `i^(-43)+i^(-32)` ?

Answer 1

`i^(-43)+i^(-32) = i+1`

Explanation:

Look at the first few non-negative powers of `i`:

`i^0 = 1`

`i^1 = i`

`i^2 = -1`

`i^3 = -i`

`i^4 = 1`

Basically this pattern: `1, i, -1, -i` repeats every `4` powers.

In terms of angles, multiplying by `i` is an anticlockwise rotation of `pi/2` in the complex plane. So after `4` rotations we are back facing the same way.

So in general we can write:

`{ (i^(4n) = 1), (i^(4n+1) = i), (i^(4n+2) = -1), (i^(4n+3) = -i) :}`

which holds for any integer `n`.

Now:

`-43 = -44+1 = 4(-11)+1`

`-32 = -32+0 = 4(-8)+0`

So:

`i^(-43)+i^(-32) = i+1`