1. what pattern does the power of i follow?
2. how can you simplify i^5689? what can be the base of your reasoning?
3. what can support your reasoning?

Step-by-step explanation:

Start by doing a much simpler problem

One

i^0 = 1

i^1 = i

i^2 = i^2 = - 1

i^3 = i^3 = (-1) * i = -i

i^4 = i^4 = (i)^2 * i^2 = -1 * -1 = 1

So now you are ready to tackle the question you were given.

Two

Every 4th power of i repeats itself. You must get one of the four answers above.

So divide 5689 by 4. What do you get?

1422.25

What does that mean?

It means that you run through the cycle of four 1422 times.  What does that mean?

1 * 1 * 1 * 1 * 1 * 1 * 1 * 1 *1

There are 1422 ones in that expression.

What about the 0.25? What about that?

That means that you have one i left over.

So the answer is i

Three

The simple question defines the pattern which will repeat itself over and over again. By knowing how the simple pattern works you can develop any power of i. For example

i^15622 =

3905.5

0.5 is the same as 2/4

i^2 = -1

i^15622 = - 1

1.

2.

Step-by-step explanation:

3. After i⁴, the cycle repeats itself from the very beginning [i = i⁵ (and so on)]. Every 4⁄4 that go by, you start from the beginning, but you keep moving up one in the exponents.

2. Since ¼ of 5689 is 1422¼, I know that it is imaginary, or iota, so moving one down would give 5688, where ¼ of that is 1422, with NOTHING left over, and that would equal to 1. You see, you have to know your multiples and divisiblity rules. The divisiblity rule of 4 states that if the LAST TWO DIGITS of an entire number are divisible by 4, then the ENTIRE number is divisible by 4. So, from this, you can tell how to answer this.

1. The pattern is in the above answer. ↑

** This can also include NEGATIVE EXPONENTS.

I hope this helps you out alot, and as always, I am joyous to assist anyone at any time.

c

-by-step explanation:

it would be c