How do you find fifth roots of $1$ $1$ ?

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Answer 1 · 2018-06-26T21:26:52

$\Rightarrow z = e^{\frac{2}{5} k π i}$ $=> z = e^(2/5 k pi i )$

$k = {0, 1, 2, 3, 4}$ $k ={0,1,2,3,4}$

We are hence find $z$ $z$ if $z = \sqrt[5]{1}$ $z = root5 1$

$\Rightarrow z^{5} = 1$ $=> z^5 = 1$

Use how $e^(2kpi i ) =1 , AA k in ZZ$

Hence $z^5 = e^(2kpi i )$

$=> z = e^(2/5 k pi i )$

For $k ={0,1,2,3,4}$ , As any other $k$ will give related solutions

Where $re^(i theta ) = rcostheta+ irsintheta$

How do you find fifth roots of 111?