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

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