Find the value of
$w^{18}$ + $w^{21}$ + $w^{- 30}$ + $w^{- 105}$ where w is complex cube root of unity

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Solution

$ω^{18} + ω^{21} + ω^{- 30} + ω^{- 105} = (ω^{3})^{6} + (ω^{3})^{7} + (ω^{3})^{- 10} + (ω^{3})^{- 35} = 1 + 1 + 1 + 1 = 4$

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