# Imaginary number -i raised to negative power

1. Feb 23, 2010

### CE Trainee

1. The problem statement, all variables and given/known data

I came across this expression in homework and for the life of me I can't figure out how this evaluates to 0: 1 - ( -i )^-4 = 1 - 1 = 0

I know that i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. I'm just not sure how to treat the negative on the i. Do I just treat i as if it were just a regular number, i.e. (-1)^4 = 1?
Or can I just say ( -i )*( -i )*( -i )*( -i ) = 1? Can anyone shed some light on this. I know it has to be painfully simple but for some reason I just can't see it.

Thanks

2. Feb 23, 2010

### xcvxcvvc

Distribute the power:
$$(-i)^4 = (-1)^4i^4$$

3. Feb 24, 2010

### HallsofIvy

Staff Emeritus
$i(-i)= -i^2= -(-1)= 1$ so i and -i are reciprocals. In particular, $(-i)^{-1}= i$ and so $(-i)^{-4}= i^4= 1$.

4. Feb 24, 2010

### snshusat161

it is very easy problem. See,

$$1 - i^{-4}$$

= $$1 - \frac {1}{i^4}$$

= $$1 - \frac {1}{1}$$

= 1 - 1

= 0