Question 700225
{{{ f(x) = x^6 - 1 }}}
{{{ 0 = x^6 - 1 }}}
{{{ x^6 = 1 }}}
There are 6 solutions, and you have 2 of them:
{{{ 1 }}}
{{{ -1 }}}
The other 4 solutions are the imaginary roots: 
{{{ root( 6,1 ) }}}
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The argand circle has real numbers plotted horizontally
and imaginary number vertically.
The 6 solutions are spaced equally on this circle
60 degrees apart starting with {{{ 1}}}:
{{{ 1/2 + (sqrt(3)/2)*i }}}
{{{ -1/2 + (sqrt(3)/2)*i }}}
{{{ -1 }}}
{{{ -1/2 - (sqrt(3)/2)*i }}}
{{{ -1/2 + (sqrt(3)/2)*i }}}
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These are just 30-60-90 triangles scaled down by 1/2
I'll check one of these:
{{{ ( -1/2 - (sqrt(3) / 2) *i)^6 }}}
{{{ ( 1/4 + (sqrt(3) / 2)*i  + (3/4)*i^2 )^3 }}}
{{{ ( -1/2 +  (sqrt(3) / 2)*i )^3 }}}
{{{ ( -1/2 +  (sqrt(3) / 2)*i )*( -1/2 - (sqrt(3) / 2) *i ) }}}
{{{ 1/4 - (3/4)*i^2 }}}
{{{ 1/4 + 3/4 = 1 }}}
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Hope this makes sense