Question

How do you simplify `i^922`?

Answer 1

`i^922 = -1`

Explanation:

If `m, n in NN` then:

`a^(m+n) = stackrel "m+n times" overbrace ((a)(a)...(a)) = stackrel "m times" overbrace ((a)(a)...(a)) stackrel "n times" overbrace ((a)(a)...(a)) = a^m*a^n`

Hence if `m, n in NN` then:

`(a^m)^n = stackrel "n times" overbrace ((a^m)(a^m)...(a^m)) = a^(stackrel "n times" overbrace (m+m+...+m)) = a^(mn)`

So:

`i^922 = i^(2*461) = (i^2)^461 = -1^(461) = (-1)^(2*230)(-1)^1`

`= ((-1)^2)^230(-1) = 1^230(-1) = -1`