Evaluate i21?

Question

Problem is i(21st power)

Choices for answer:

i
-1
-i
1

Answer 1

The answer is i.

First, look at i^0(i to the 0th power). It is 1, because anything to the 0th power is 1.
Now, look at i^1(i to the 1st power). It is simply i.
Next, look at i^2, which is -1, by the definition of i.
i^3 = i^2 * i, right, which is -1 * i = -i.
i^4 = 1 etc..

Observe i^n, where n is any value over 3. We can then write i^n as i^4 * i^(n-4) = 1*i^(n-4) = i^(n-4). So i^n = i^(n-4). So we can just keep subtracting 4 from n until we are at 0,1,2 or 3, which we now know! We see that this number (0,1,2 or 3) is simply the remainder when we divide n by 4.

So to find powers of i, we simply find the remainder when it is divided by 4, and pit it against this chart we have made:

0:1
1: i
2: -1
3: -i

So for this problem, if we do the long division, we see that 21/4 = 5 + 1/4, so the remainder is 1. Therefore, the answer to this particular problem is i.

So the answer is i.

Answer 2

Remember, i is the square root of -1. So i^2 is -1, i^3 is -i, i^4 is 1, and i^5 is i again. It cycles through every four powers, so i^21 = i^17 = i^13 = i^9 = i^5 = i^1 = i. C.

Answer 3

Firstly,
i = √ -1

That means,
i^2 = -1 and i^4 = 1

So,
i^21 = i^4 * i^4 * i^4 * i^4 * i^4 * i^1

Substituting i^4 = 1,
i^21 = 1 * 1 * 1 * 1 * 1 * i

And you'll get
i^21 = i

Hope that helps. =)

Answer 4

i is 1 /__90o (pi/2)

So, i^21 is 1^21/__90x21

1^21 = 1

And 90x21 = 360 x 5 + 90

So, the result will be equals to 1/_90, that is: i

Ana

Answer 5

i^(21 mod 4) = i^1 = i


Doug

Answer 6

i = -1
So i^4 = (i²)²
= (-1)²
= 1

Now i^21
= i * i^20
= i * (i^4)^5
= i * 1^5
= i

Answer 7

Everyones already explained the answer...I'm just here to tell you that you can plug it in your calculator... much easier..

Answer 8

I, C?

Answer 9

( i )....good luck