Let us find the six roots of the equation z6=−1 using de Moivre's Theorem for fractional power:
[r(cosθ+isinθ)]n1=nr(cosnθ+2πk+isinnθ+2πk) for k∈{0,1,...,n−1}.
In trigonometric form −1=cosπ+isinπ, that is r=1,θ=π. Consequently, the formula
zk=cos6π+2πk+isin6π+2πk=e6π+2πki for k∈{0,1,2,3,4,5} gives all six different roots.
Therefore, we have:
z0=e6πi, z1=e63πi=e2πi, z2=e65πi, z3=e67πi, z4=e69πi=e23πi, z5=e611πi.
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