Question

What is ${\omega }^{100}+{\omega }^{200}+{\omega }^{300}$ equal to, where $\omega$ is the cube root of unity?

• A. 1
• B. $3\omega$
• C. $3{\omega }^2$
• D. $0$

Answer 1

The correct option is D $0$

$w$ is the cube root of unity.

$\therefore w^3=1$ ...(1)

and $1+w+w^2=0$ ...(2)

$w^{100}+w^{200}+w^{300}$

$=\left(w^{99}\right)\left(w\right)+\left(w^{198}\right)\left(w^2\right)+\left(w^{300}\right)$

$=\left(w^3{)}^{33}\right(w)+(w^3{)}^{66}\left(w^2\right)+(w^3{)}^{100}$

$=\left(1{)}^{33}\right(w)+(1{)}^{66}\left(w^2\right)+(1{)}^{100}$ ...(using 1 )

$=w+w^2+1$

$=0$ ...(using 2)

So, the answer is option (D)