First slide

De-moivre's theorem

Question

If $α (\neq 1)$ is a fifth root of unity and $β (\neq 1)$ is a fourth root of unity then
$z = (1 + α) (1 + β) (1 + α^{2}) (1 + β^{2}) (1 + α^{3}) (1 + β^{3})$ equals

Moderate

A

$α$
B

$β$
C

$α β$
D

0

Solution

As $β \neq 1$ is a fourth root of unity,
$β^{4} = 1 \Rightarrow (1 - β) (1 + β + β^{2} + β^{3}) = 0$
As $β \neq 1, 1 + β + β^{2} (1 + β) = 0$
$\begin{array}{ll} \Rightarrow (1 + β) (1 + β^{2}) = 0 \\ ∴ z = 0 \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App