1. ## complex numbers

not sure if this is the right forum so sorry if i got it wrong. i'm having alot of trouble with this question. the numbers in brackets in front of the w's are meant to be subscripts

if $\displaystyle w_n = \exp\left(2\pi i/n\right)$ show $\displaystyle w_n^n = 1$ and that

$\displaystyle 1 + w_n +w_n^2 + w_n^3 + \dots + w_n^{n-1} = 0$

and show

$\displaystyle \left(x + y\cdot w_3 + z\cdot w_3^2\right)\left(x + y\cdot w_3^2 + z\cdot w_3\right) = x^2 + y^2 +z^2 - xy -yz -zx$

sorry for the lack of tex in it. thanks so much for any help given!!

------------------------------------------------------------

Edit by Chris L T521: Reformatted question with $\displaystyle \text{\LaTeX}$. Please inform me if I have misinterpreted anything.

2. $\displaystyle w^k_{n}=e^{\frac{2k\pi}{n}}=\left(e^{\frac{2\pi}{n }}\right)^k$

If $\displaystyle k=n$,then,$\displaystyle w^n_{n}=\cos 2\pi+i\sin 2\pi=1+i.0=1$

$\displaystyle 1+w_{n}+w^2_{n}+w^3_{n}+.....+w^{n-1}_{n}=\frac{1-w^n_{n}}{1-w_{n}}=\frac{1-1}{1-w_{n}}=0$

For,$\displaystyle n=3$,anove results are,$\displaystyle w^3_{3}=1$ and $\displaystyle 1+w_{3}+w^2_{3}=0$

$\displaystyle x^2+y^2w^3_{3}+z^2_{3}+xy(w_{3}+w^2_{3})+yz(w^2_{3 }+w^4_{3})+zx((w_{3}+w^2_{3})$

$\displaystyle =x^2+y^2+z^2+xy(-1)+yz(w^2_{3}+w_{3})+zx(-1)$

$\displaystyle =x^2+y^2+z^2-xy-yz-zx$