Simplify i22. possible choices are 1, -1 ,i ,-i

I assume you mean i^22

There’s a cycle i follows when being taken to a certain power, so let’s see what that is:

i^1 = i
i^2 = -1
i^3 = - i
i^4 = 1

And i^5 = i, so our cycle repeats after every 4th power. To find i^22 then, we can divide 22 by 4 to obtain an answer of 5, with a remainder of 2. In other words, 22 = 4(5) + 2

So, i^22 = i^[4(5) + 2]

We can break that up into the product i^[4(5)] x i^2

We know that i^4 is 1, and 1^5 is still 1, so i^[4(5)] = 1; i^2 is simply -1, so altogether, we find that

i^[4(5)] x i^2 = 1 x (-1) = -1

Our answer then is -1.