The smallest positive integer `n` for which `((1+i)/(1-i))^n=i` is 8 (b) 16 (c) 12 (d) None of these

Options

A. 1
B. 4
C. 6
D. 16

Answer

B

Explanation

`((1+i)/(1-i))^(n)=1`
We know,`i^(2) =-1 and i^(4)=1.`
Now, rationalise the denominator.
`(((1+i)(1+i))/((1-i)(1+i)))^(n)=((1+2i+i^(2))/(1-i^(2)))=((1+2i-i)/(1-(-1)))^(n)=((2i)/(2))^(n)=-i^(n)`
The smallest prositive integer for which `i^(n)=1 is 4`