Question 1160400
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Let me start teaching you how to formulate this problem CORRECTLY.


<pre>
    the number  " 1 "  {{{highlight(IS)}}} the root of the equation w^5 = 1.

    Therefore, there is <U>no need</U> to write  "if 1 and w are two of the five roots of w^=1.



             (otherwise, it makes very strange impression).




    The correct writing is "if w is the root of the equation w^5 = 1, different from 1 . . . and so on . . . ".
</pre>


Now to the solution.


<pre>
If w is the root of the equation w^5 = 1, then


    {{{(w^2)^5}}} = {{{w^(2*5)}}} = {{{(W^5)^2}}} = {{{1^2}}} = 1,

    so {{{w^2}}}  is the root of the same equation.



Same proof works for  {{{w^3}}}.


    {{{(w^3)^5}}} = {{{w^(3*5)}}} = {{{(W^5)^3}}} = {{{1^3}}} = 1,

    so {{{w^3}}}  is the root of the same equation.



Same proof works for  {{{w^4}}}.

Do it on your own, as a useful exercise.
</pre>

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Proved and solved. And completed.