If sum of the cube roots of unity is 0 Then prove that Product of cube roots of unity =1 Anyone?

1 Answer
MattyMatty
Feb 8, 2018

#"See explanation"#

Explanation:

#z^3 - 1 = 0" is the equation that yields the cube roots of"#
#"unity. So we can apply the theory of polynomials to"#
#"conclude that "z_1*z_2*z_3 = 1" (Newton's identities)."#

#"If you really want to calculate it and check it :"#
#z^3 - 1 = (z - 1)(z^2 + z + 1) = 0#
#=> z = 1 " OR " z^2+z+1 = 0#
#=> z = 1 " OR " z = (-1 pm sqrt(3) i)/2#
#=> (z_1)*(z_2)*(z_3) = 1*((-1+sqrt(3) i)/2)*(-1-sqrt(3) i)/2#

#= 1*(1+3)/4 = 1#

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