For what m is the complex number $(\sqrt 3+i)^m$ positive and real?

[Santilopez10]

Homework Statement

Find all $$n \in Z$$, for which $$ (\sqrt 3+i)^n = 2^{n-1} (-1+\sqrt 3 i)$$

Homework Equations

$$ (a+b i)^n = |a+b i|^n e^{i n (\theta + 2 \pi k)} $$

The Attempt at a Solution

First I convert everything to it`s complex exponential form: $$ 2^n e^{i n (\frac {\pi}{3}+ 2\pi k)} = 2^{n-1} 2 e^{i (\frac{2 \pi}{3} +2 \pi k)} $$
this simplifies to $$ e^{i n (\frac {\pi}{3}+ 2\pi k)} = e^{i (\frac{2 \pi}{3} +2 \pi k)} $$
I know how to find an expression for n, but not that it`s only in the field of $Z$, any help would be appreciated, thanks!

 

[Orodruin]

Hint: There is no point to writing out the ks. Instead, try to figure out when the numbers are equal without using them. Also, if you do use them, you cannot assume that they are the same on both sides.

 

[Santilopez10]

Well I would get $$ n \frac {\pi}{3} = \frac {2 \pi}{3} $$
which is true for n = 2, n =8, n= 14, n= 20 ... 2+6t if I am not wrong, but this still does not satisfy the answer provided by the book which is n= 4(1+3t) I am missing a 2!

 

[SammyS]

Well I would get $$ n \frac {\pi}{3} = \frac {2 \pi}{3} $$
which is true for n = 2, n =8, n= 14, n= 20 ... 2+6t if I am not wrong, but this still does not satisfy the answer provided by the book which is n= 4(1+3t) I am missing a 2!

Actually that equation is only true for n = 2, but that's beside the point.Your primary difficulty lies in the fact that the $\ \dfrac \pi 3 \$ is incorrect.

$\sqrt{3} + i \ne 2 e^{(\pi/3)i}$

Rather: $\ 2 e^{(\pi/3)i} = 1 +i\sqrt{3} \,.$

 

[Santilopez10]

Actually that equation is only true for n = 2, but that's beside the point.Your primary difficulty lies in the fact that the $\ \dfrac \pi 3 \$ is incorrect.

$\sqrt{3} + i \ne 2 e^{(\pi/3)i}$

Rather: $\ 2 e^{(\pi/3)i} = 1 +i\sqrt{3} \,.$

Thanks, now I got the correct answer. By the way, when I mentioned the various answers for n it was in the context of angles, not numbers a s a whole.

 

[lurflurf]

I would rewrite the right hand side
$$(\sqrt 3+i)^n = 2^{n-1} (-1+\sqrt 3 i)$$
$$(\sqrt 3+i)^n = 2^{n-4}(\sqrt 3+i)^4$$
for what m is
$$(\sqrt 3+i)^m$$
positive and real?