Simplify? :) help

Question

Simplify? :) help

anonymous · Answer

$$\prime77$$
it's supposed to be (i^77) if you cant tell.
I know that (i) is equal to 1. so would it be 77*1=77?

hero · Answer

Actually $i = \sqrt{-1}$

hero · Answer

Ashley, I help many students with questions and I cannot respond to your personal messages since you have blocked me.

hero · Answer

So what you have is 

$(\sqrt{-1})^{77}$

anonymous · Answer

ok

precal · Answer

start with an easier pattern and then build

anonymous · Answer

do i just multiply -1 *77?

hero · Answer

Actually, the easiest way is to do 77/4 and get a remainder of 1. Then you know the answer is equal to $i$

precal · Answer

yes, Hero is correct.

anonymous · Answer

77/ 4 is my answer? 15.4?

precal · Answer

no answer is "i"

hero · Answer

Hmmm, not exactly. The whole point of dividing is to get the remainder. The remainder you get becomes the new exponent

hero · Answer

Since in this case, the remainder of 77/4 is one, your new exponent becomes

$\sqrt{-1}^1$

anonymous · Answer

im really confused lol
I start with 
sqrt -1 ^77
then i divide 77/4 and get 15.4
then what?

hero · Answer

So $(\sqrt{-1})^1 = \sqrt{-1} = i$

hero · Answer

Well, actually, you're supposed to 

do 77/4 and get 19 remainder 1

anonymous · Answer

so what happened to the 77 and the 19?

dumbcow · Answer

$$i^{2} = -1$$
$$\rightarrow i^{77} = (i^{2})^{38}*i = (-1)^{38}*i = i$$

hero · Answer

4 x 19 = 76
76 + 1 = 77

dumbcow · Answer

@LaurenAshley1201 
there are only 4 possible answers for i^n
1, -1, i, -i

hero · Answer

@dumbcow provides an alternate way, but I thought that maybe you would prefer to simply divide to find the remainder since it is a simple and more direct process.