The powers of i is always equal to either one of these 4 numbers: 1, i , -1,-i. Here we will see why.

i^{0} = 1, this is because for any any non-zero number, any number with exponent 0 is equal to 1.

i^{1} = i, this is because any number with exponent 1 is equal to itself, in this case i itself.

i^{2} = -1, this can be explained by the definition of imaginary number, which says i is defined as `sqrt(-1), so its square is -1.`

i^{3} = -i, since i^{2}=-1, i^{3} = i^{2} * i = -1 * i = -i

i^{4} = 1 becausei^{4} = i^{3} * i = -i * i = -`sqrt(-1)` ` `*`sqrt(-1)` = 1

We see that i^{4 }= i^{0}, so this is a cycle, so by prediction i^{5} equals to i, i^{6} equals -1, and so on.... and in fact that is true.

So i with the exponent n (any whole number from 0 to above) can be determined by the following rules:

Divide the exponent by 4 and find the remainder,

if the remainder is 0 (no remainder), then i^{n}_{ }equals 1;

if the remainder is 1, then i^{n} equals i;

if the remainder is 2, then i^{n} equals -1;

if the remainder is 3, then i^{n} equals -i.

**Subject :**Math**Topic :**Algebra 2-
**Posted By :**Jason **Created on :**11-29-2012

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