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If $\alpha$ is the fifth root of unity, then the value of $log_2 |1+\alpha+\alpha^2+\alpha^3-\large\frac{1}{\alpha}$$|=?$

$\begin{array}{1 1} -1 \\ 0 \\ 1 \\ 2 \end{array} $

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Toolbox:
  • Sum of all $n^{th}$ roots of unity = $0$
  • $log_a a=1$
Given: $\alpha$ is the fifth root of unity.
$\Rightarrow\:\alpha^5=1$, $ |\alpha|=1$ and $1+\alpha+\alpha^2+\alpha^3+\alpha^4=0$
$\Rightarrow\:\alpha^4\alpha=1$
$\Rightarrow\:\alpha^4=\large\frac{1}{\alpha}$
$\Rightarrow\:|1+\alpha+\alpha^2+\alpha^3-\large\frac{1}{\alpha}|$$=|1+\alpha+\alpha^2+\alpha^3-\alpha^4|$
$=|-2\alpha^4|=2|\alpha|^4=2$
$\Rightarrow\:log_2 |1+\alpha+\alpha^2+\alpha^3-\large\frac{1}{\alpha}|$$=log_2 2=1$
answered Aug 6, 2013 by rvidyagovindarajan_1
 

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