1. Tardigrade
  2. Question
  3. Mathematics
  4. 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 Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. Example 15: If $\alpha(\neq 1)$ is a fifth root of unity and $\beta(\neq 1)$ is a fourth root of unity then
$z=(1+\alpha)(1+\beta)\left(1+\alpha^2\right)\left(1+\beta^2\right)\left(1+\alpha^3\right)\left(1+\beta^3\right)$ equals

Complex Numbers and Quadratic Equations

Solution:

As $\beta \neq 1$ is a fourth root of unity,
$\beta^4=1 \Rightarrow(1-\beta)\left(1+\beta+\beta^2+\beta^3\right)=0$
As $ \beta \neq 1,1+\beta+\beta^2(1+\beta)=0$
$\Rightarrow (1+\beta)\left(1+\beta^2\right)=0 $
$\therefore z=0$