what is i^14?

Question

imaginary numbers...help!

Answer 1

if u know what an argand diagram looks like u can use this to work out i^n where n equals a positive whole number.
starting at the positive imaginary or y-axis for i and then moving anticlockwise 90 degrees for each time u add 1 to the power.

Since 14 is an even number u can take 2 out of the power:
i.e.

i^14=(i^2)^7.

But, i^2=-1, so your problem becomes:

-1^7.

This is just multiplying -1 by itself 7 times which gives -1 still

-1 raised to an odd power is always -1, so -1^7=-1.

so u end up with: i^14= -1

the way i think of it is to divide the power by 4,
- if the remainder is 0, the answer is 1 (i^0).
- if the remainder is 1, the answer is i (i^1).
-if the remainder is 2, the answer is -1 (i^2).
-if the remainder is 3, the answer is -i (i^3).

Answer 2

Hi Hippy Chick,

There are a couple of ways to solve the problem. Consider the diagram

| y (i -- imaginary)
|
|
|
|
----------------------------0------------------- x (real)
|
|
|
|
|

Such a diagram is called an Argand Diagram (unfortunately, Yahoo does not lend itself very well to drawing diagrams or answering mathematical questions). Now, if we were to plot i, then i would fall right on the upper y - axis. We know that i * i = -1. If we plot -1, then -1 falls on the left x-axis. Multiplying through by i again, and the answer, -i, falls on the lower y axis. We know that i * (-i) = 1, since (i*i) * (i * i) = 1.

So, we note this important fact: multiplying i by itself 4 times gives 1, and thus any number of multiples of 4, such as 4, 8, 12, ... 200, 4000, etc, gives the same answer: 1.

So, to solve our problem, simply note that 4 * 3 = 12, and 14 / 4 = 12 remainder 2. So, we're left with that remainder 2, which gives us the number of times we have to multiply i by itself to finally get the answer. i * i = -1, and 1 * (-1) = -1. That is the answer.

Typing is not my strong point (I got a D in typing in HS). I think I will stop now. But I do hope this helps.

James :-)

Answer 3

For info, i^4 = 1, i^2 = -1.
i^14 = i^4 * i^4 * i^4 * i^2 = 1 * 1 * 1 * -1 = -1.
Easy, right?

Answer 4

14 mod 4 = 2

i^14=i^2 = -1

Answer 5

Well, i^4 = 1, so i ^12 =(i^4)³ = 1.
i ^14 = i^12 *i² = i² = -1.
Answer: -1.

Answer 6

-1

Answer 7

(i^2)^7 = -1^7 = -1

Answer 8

i = sqrt(-1)

so i^2 = -1

now i^14 is the same as (i^2)^7 => (-1)^7 = -1 * -1 * -1 *-1*-1*-1*-1 = -1

so i^14 = -1