Question

For what minimum value of n is [(1+I)/(1-i)]ⁿ real?

Answer 1

[(1+i)/(1-i)]ⁿ

First, we rationalize the given expression by multiplying and dividing the given expression by the conjugate of denominator,

[(1+i)/(1-i) . (1+i)/(1+i)]ⁿ = x (say)

This means (2i/2)ⁿ = x

This gives iⁿ = x

As, i⁻² = −1. i⁻¹ = −i, i⁰ = 1. i¹ = i., i² = −1. i³ = −i. i⁴ = 1. i⁵ = i. i⁶ = −1. iⁿ = i..

therefore the minimum value of n is -2 for which iⁿ = x is real