Question

If $w$ is the complex cube root of unity, find the value of
i) $w+\frac{1}{w}$
ii) $w^2+w^3+w^4$
iii) $(1+w^2{)}^3$
iv) $(1-w-w^2{)}^3+(1-w+w^2{)}^3$

Answer 1

Given that $w$ is the complex cube root of unity

We know that $1+w+w^2=0$ .... $\left(1\right)$

and $w^3=1$ ...... $\left(2\right)$

i) $w+\frac{1}{w}=\frac{1+w^2}{w}=-\frac{w}{w}=-1$ ..... (from (1))

ii) $w^2+w^3+w^4=w^2(1+w+w^2)=0$ ..... (from (1))

iii) $(1+w^2{)}^3=(-w{)}^3=-w^3=-1$ ..... (from (1) and (2))

iv) $(1-w-w^2{)}^3+(1-w+w^2{)}^3=(1+1{)}^3+(-w-w{)}^3=1-(w^3{)}^2=1-1=0$ (from (1) and (2))