Question

State whether the following statement is true or false.

If $\omega$ is one of the complex cube roots of unity, then
$(1-\omega +{\omega }^2{)}^3=(1+\omega -{\omega }^2{)}^3=-8$

• A. True
• B. False

Answer 1

The correct option is A True

We know that if $w$ is complex cube root of unity then $(1+w+w^2)=0$ and $w^3=1$..........[1]

So, $(1+w^2)=-w$ and $(1+w=-w^2)$..........[from 1].........[2]

Now,

$(1+w^2-w{)}^3=(-w-w{)}^3=(-2w{)}^3=(-2{)}^3(w{)}^3=-8(1)=-8$

Also,

$(1-w^2+w{)}^3=(-w^2-w^2{)}^3=(-2w^2{)}^3=(-2{)}^3(w^2{)}^3=-8(w^6)=-8$

So,

$(1-w+w^2{)}^3=(1+w-w^2{)}^3=-8$