Determine-smallest-n-0-for-which-i-n-1-

Question Number 8336 by Rasheed Soomro last updated on 08/Oct/16

Determine smallest n (\neq 0), for which

{(ω + i)}^{n} = 1 .

Commented by prakash jain last updated on 08/Oct/16

w = \cos \frac{2 π}{3} + i \sin \frac{2 π}{3} = - \frac{1}{2} + i \frac{\sqrt{3}}{2}

w + i = - \frac{1}{2} + i (\frac{\sqrt{3}}{2} + 1)

∣ w + i ∣= \sqrt{{(\frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2} + 1)}^{2}}

= \sqrt{\frac{1}{4} + 1 + \frac{3}{4 + 2 \sqrt{3}}} = \sqrt{2 + \sqrt{3}}

Since ∣ w + i ∣= \sqrt{2 + \sqrt{3}}

of {(w + i)}^{n} = 1 \Rightarrow n = 0

There is no value of n \neq 0 for which

{(w + i)}^{n} = 1

Commented by Rasheed Soomro last updated on 09/Oct/16

Thanks a lot!