complex number i to the 25th power.?

Question

I know you divide it by 4 but i forgot what is the answer when you get the remainders what are the answers. i tried to read the book but couldn't understand. thanks

Answer 1

so you already know about dividing by 4
since
i x i = -1
i x i x i x i = 1
i^4 = 1

25 / 4 = 6 remainder of 1 translates to
i^25 = (i^4)^6 * i^1
remainder........^
answer is i

Answer 2

i^0=1
i^1=i
i^2=-1
i^3=-i
i^4=1
...

So you are basically taking the remainder when the exponent is divided by four. If it is zero, then the answer is 1. If it is 1, the answer is i. If it is two, the answer is -1. If it is 3, the answer is -i. It follows that i^25=i

Answer 3

xpress -2 in polar form = 2

Answer 4

i=i
i^2=i*i=-1
i^3=i^2*i=-i
i^4=i^3*i=1
i^5=i^4*i=i
and so on. Your answer is i (the remainder is 1, so you got i^1=i)

Answer 5

i^1 is just i

i^2 is -1 since i is the square root of -1

i^3, which is i^2 times i, is -1i or -i

i^4, which is i^2 times i^2 , is (-1)(-1) or 1

Here it repeats: i^5 = i^4 times i^1 =1 times i = i

i^6 = i^4 times i^2 = 1 (-1) = -1
etc

So if your remainder is 1, you get i

If it is 2, you get -1

If it is 3, you get -i

If it is 0, you get 1

Answer 6

The key is to use mod 4. Each time you multiply i by itself, you in effect rotate it thru 90 degrees on the unit circle. So i^2 = -1 because you've rotated it thru 2x90 degrees and you're at (-1,0) on the unit circle. i^3 = -i because you've rotated it 270 degrees and you're at (0,-i) on the unit circle. Then with i^4 yo'ure back at 1 or (1,0) on the unit circle.
There's lots of good info online about this, see for example
http://www.math.rutgers.edu/~erowland/modulararithmetic.html

Answer 7

i^2 = -1
(-1)^2 = 1
therefore i^4 = 1.

You have:
i^25 = (i^1)*(i^24)
= i*((i^4)^6)
= i*(1^6)
= i