Question

If $1,{\alpha }_1,{\alpha }_2,{\alpha }_3$ are the fourth roots of unity, then the value of $\left(1+{\alpha }_1\right)\left(1+{\alpha }_2\right)\left(1+{\alpha }_3\right)$ is equal to

• A. $-3$
• B. $-1$
• C. $0$
• D. $2$

Answer 1

The correct option is C $0$

Let $1,{\alpha }_1,{\alpha }_2,{\alpha }_3$ are the fourth rotts of unity.

We know that the fourth roots of unity are $1,i,-1,-i$

All these roots are got by solving equation $x=(1{)}^{\frac{1}{4}}$

By using demovire's theorem.

Now,

$(1+{\alpha }_1)(1+{\alpha }_2)(1+{\alpha }_3)$

$\Rightarrow$ $(1+i)(1+(-1\left)\right)(1+(-i\left)\right)$

$\Rightarrow$ $(1+i)\left(0\right)(1-i)$

$\Rightarrow$ $0$