Question

If $z^2-z+1=0$ then $z^n-z^{-n}$, where $n$ is a multiple of $3$, is

• A. $2(-1{)}^n$
• B. $0$
• C. $(-1{)}^{n+1}$
• D. none of these

Answer 1

The correct option is B $0$

$z^2-z+1=0$

Here, we can say that

z is cube root of unity

$z=\omega ,{\omega }^2$

Considering $z=\omega$

$z^n-z^{-n}={\omega }^n-{\omega }^{-n}$

Now ${\omega }^n=1$ when $n=3k$

${\omega }^n=1,{\omega }^{-n}=\frac{1}{{\omega }^n}=\frac{1}{1}=1$

${\omega }^n-{\omega }^{-n}=0$

$z^n-z^{-n}=0$