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For a positive integer n , find the valu...

For a positive integer `n` , find the value of (`1-i)^n(1-1/i)^ndot`

Text Solution

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Given expression = `(1-i)^(n)(1-(1)/(i))^(n)`
`=(1 -i)^(n)(i-1)^(n).i^(-n)= (1-i)^(n)(-1)^(n). i^(-n)`
`=[(1 -i)^(2)]^(n)(-1)^(n).i^(-n)= (1-i^(2)-2i)^(n)(-n)^(n) i^(-n)" "[:.i^(2)=-1]`
`=(1-1-2i)^(n)i^(-n)=(-2)^(n). i^(n)(-1)^(n)i^(-n) `
`= (-1)^(2n).2^(n) = 2^(n)`
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