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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}$$|=? \begin{array}{1 1} -1 \\ 0 \\ 1 \\ 2 \end{array} Can you answer this question? ## 1 Answer 0 votes Toolbox: • Sum of all n^{th} roots of unity = 0 • log_a a=1 Given: \alpha is the fifth root of unity. \Rightarrow\:\alpha^5=1, |\alpha|=1 and 1+\alpha+\alpha^2+\alpha^3+\alpha^4=0 \Rightarrow\:\alpha^4\alpha=1 \Rightarrow\:\alpha^4=\large\frac{1}{\alpha} \Rightarrow\:|1+\alpha+\alpha^2+\alpha^3-\large\frac{1}{\alpha}|$$=|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