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>> Cube Root of Unity Part-1
Cube Root of Unity
(i) The cube roots of unity are 1, (-1+i√3)/2, (-1-i√3)/2 .
(ii) If w is one of the imaginary cube roots of unity then 1+w+w2=0. In general 1 + wn+w2n=0; where n ε I but is not the multiple of 3.
(iii) In polar form the cube roots of unity are:
cos 0 + i sin 0; cos2Π/3 + isin2Π/3; cos4Π/3 + isin4Π/3.
(iv) The three cube roots of unity when plotted on the argand plane constitutes the vertices of an equilateral triangle.
(v) The following factorization should be remembered:
(a, b, c ε R and w is the cube root of unity)
a3 - b3= (a-b) (a-wb) (a-x2b);
a3 + b3= (a+b) (a+wb) (a+x2b);
a3 + b3 + c3 - 3abc = (a+b+c) (a+wb+w2c) (a+w2b+wc);
nth Roots of unity -
if 1, α1, α2,......αn-1 are the n, nth roots of unity then: