Question

If 1,$\alpha$,${\alpha }^2$,${\alpha }^3$,............,${\alpha }^{n-1}$ are roots of unity.What is the value of If 1 $\times$ $\alpha$ $\times$ ${\alpha }^2$ $\times$ ${\alpha }^3$, $\times$ ............,${\alpha }^{n-1}$.

• A. $(-1{)}^{n+1}$
• B. $(-1{)}^n$
• C. $(-1{)}^{n-1}$
• D. $(-1{)}^{n+1}$

Answer 1

The correct option is C

$(-1{)}^{n-1}$

If 1,$\alpha$,${\alpha }^2$,${\alpha }^3$,............,${\alpha }^{n-1}$ are the roots of unity.

Then 1 $\times$ $\alpha$ $\times$ ${\alpha }^2$ $\times$ ${\alpha }^3$, $\times$ ............,${\alpha }^{n-1}$.

= ${\alpha }^{1+2+3+.....(n-1)}$

=$\alpha \frac{(n-1)(n-1+1)}{2}$

=$\alpha \frac{n(n-1)}{2}$

= $(cos\frac{2\pi }{n}+isin\frac{2\pi }{n}{)}^{\frac{n(n-1)}{2}}$

= $cos\pi (n-1)+isin\pi (n-1)$

= $(cos\pi +isin\pi {)}^{n-1}$

=$(-1{)}^{n-1}$