Imaginary number -i raised to negative power

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  • #1
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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
 

Answers and Replies

  • #2
393
0

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:
[tex](-i)^4 = (-1)^4i^4[/tex]
 
  • #3
HallsofIvy
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[itex]i(-i)= -i^2= -(-1)= 1[/itex] so i and -i are reciprocals. In particular, [itex](-i)^{-1}= i[/itex] and so [itex](-i)^{-4}= i^4= 1[/itex].
 
  • #4
214
1
it is very easy problem. See,

[tex]

1 - i^{-4}[/tex]

= [tex]1 - \frac {1}{i^4}[/tex]

= [tex]1 - \frac {1}{1}[/tex]

= 1 - 1

= 0
 

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