Question

What is the smallest positive integer n for which (1 + i)²ⁿ = (1 – i)²ⁿ?

Answer 1

Given: 

⇒ (1+i)²ⁿ=(1-i)²ⁿ 

⇒ ((1+i)²)ⁿ=((1-i)²)ⁿ 

⇒ (1²+i²+2(1)(i))ⁿ=(1²+i²-2(1)(i))ⁿ 

We know that i²=-1 

⇒ (1-1+2i)ⁿ=(1-1-2i)ⁿ 

⇒ (2i)ⁿ=(-2i)ⁿ 

We can see that the Relation holds only when n is an even integer. 

∴ The smallest positive integer n is 2.