Simplify this expression:
i ^257

Okay! 
What is i? 
i = √(-1)

What is i^2? 
i^2 = √(-1) * √(-1) 

Note: Whenever you multiply two square roots with the same numbers you get this: 

√(a^2)

Or in this case: 

√(-1 * -1) = -√(1) = -1

So, 
i^2 = -1

What does i^3 equal? 
i^3 = √(-1) * √(-1) * √(-1) = 1 * √(-1) = √(-1) which is... -i

What does i^4 equal? 
i^4 = √(-1) * √(-1) * √(-1) * √(-1) =  √(-1 * -1 * -1 * -1) = 1

After that the cycle does it thing and cycles back. 
So, with this information we can figure out what i^157 equals. 

Knowing long division here helps a lot!
Take 157 and divide it by 4 using long division. If you don't know it, I'll show you how to do it to the best of my abilities. (Refer to the picture. If you have any questions about it then ask. Even if the question is... what does this say?) 

After using long division we see that we get: 
39 with a remainder of 1. 

What that means is: 
(i^4)^39 is what we have. The remainder of 1 means that we have 1 i left. So our answer is i. 

The reason we divided by 4 and not some other term is because i^4 = 1, so it is easy to work with. In this case (i^4)^39 still just equals 1. Which leaves our remainder as our answer. 

So once again the final answer is: 
i

As for i^257. I'll show you the answer, but I won't give the work for this one. I'd recommend you try it out for yourself. 

i^257 = (i^4)^64 * i = i.

I hope that I helped, and if you have any further questions then ask away! 

The graph isn’t that clear