Imaginary number -i raised to negative power

[CE Trainee]

Homework Statement

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

 

[xcvxcvvc]

Homework Statement

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

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

 

[HallsofIvy]

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

 

[snshusat161]

it is very easy problem. See,

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

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

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

= 1 - 1

= 0