Question

The smallest positive integer n for which

(1+√3i)^n/2 is real is​

Answer 1

Answer:

It is given that, [(1 + i√3)/(1 - i√3)]ⁿ = 1

first of all, resolve (1 + i√3)/(1 - i√3)

(1 + i√3)(1 + i√3)/(1 - i√3)(1 + i√3)

= (1 + i√3)²/(1² - i²√3²)

= (1 + i²√3² + 2i√3)/(1 + 3)

= (1 - 3 + 2i√3)/(1 + 3)

= (-1 + i√3)/2

= (-1/2 + i√3/2)

= sin(-π/6) + icos(-π/6)

now, [(1 + i√3)/(1 - i√3)]ⁿ = {sin(-π/6) + icos(-π/6)}ⁿ

= sin(-nπ/6) + i cos(-nπ/6)

now, if choose n in such a way that, sin(-nπ/6) + i cos(-nπ/6) becomes 1

so, n = 9 is answer .

PLEASE MARK THIS ANSWER AS BRAINLIEST ANSWER