Question

How do you simplify `i^656`?

Answer 1

`i^656 = 1`

Explanation:

We will use the following:

• `i^2 = -1 => (i^2)^2 = (-1)^2 = 1`

• `(x^a)^b = x^(ab)`

As `656 = 4*164` this means

`i^656 = i^(4*164) = (i^4)^164 = 1^164 = 1`

In general, it is easy to evaluate `i^n` by "pulling out" the greatest multiple of `4` possible from the exponent. Either the exponent will be a multiple of `4`, as above, or we will be left with `i`, `i^2`, or `i^3`. For example:

`i^23 = i^20i^3 = i^(4*5)i^3 = (i^4)^5i^3 = i^3 = i^2i = (-1)i = -i`