Example 15: If α(≠ 1) is a fifth root of unity and β(≠ 1) is a fourth root of unity then z=(1+α)(1+β)(1+α2)(1+β2)(1+α3)(1+β3) equals

Question

Example 15: If α(≠ 1) is a fifth root of unity and β(≠ 1) is a fourth root of unity then z=(1+α)(1+β)(1+α2)(1+β2)(1+α3)(1+β3) equals (A) α (B) β (C) α β (D) 0

Tardigrade Admin · Accepted Answer

As β ≠ 1 is a fourth root of unity, β4=1 ⇒(1-β)(1+β+β2+β3)=0 As β ≠ 1,1+β+β2(1+β)=0 ⇒ (1+β)(1+β2)=0 ∴ z=0

Q. Example 15: 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

$α$

$β$

$α β$

0

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

Q. Example 15: If α(=1) is a fifth root of unity and β(=1) is a fourth root of unity then z=(1+α)(1+β)(1+α2)(1+β2)(1+α3)(1+β3) equals

α

β

αβ

0

As β=1 is a fourth root of unity, β4=1⇒(1−β)(1+β+β2+β3)=0 As β=1,1+β+β2(1+β)=0 ⇒(1+β)(1+β2)=0 ∴z=0

Q. Example 15: 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

$α$

$β$

$α β$

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