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1.7 Cube roots of unity : Number 1 is often called unity, , , , , , , , , , Let x be the cube root of unity ie. 1 & =, . x8-1=0, 6 (x-1D@?+x+1)=0 ox—-1=0 orx*+x+1=0, aes 1 ores 14+ (1)2=4x1x1, 2x1, “x=1orx= Es, “x=1orx= ates:, .x=1, x= ist i “= Ne are cube roots of unity, , 2, Let w be the complex cube root of unity ,, 1! i -1-V3i, , wit = wthen 22 — w?, Hence cube roots of unity are 1 ,w, w?., , am, 4m,, ,w=e3 ,w*=e3, , Let

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Let w be the complex cube root of unity 1., , - w is complex root of eq. x? +x+1=0and x? =, (i) w* +w+1=0, , (ii) w? =, , Power of w, , (iii) w3" = 1, , (iv) wet =w, , (vi) mert2 = q2

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e 2) Ifw isa complex cube root of unity, show that, (i) 2- w)(2- w*) =7, , Pe (i) SOLUTION:- let w isa complex cube root of unity., w*ta+1=0 & w=, , w* +m = -1, , , , e LHS.= (2— w)(2- w”), = 4-20? —-2w+w?, = 4- 2(w*+w) + w?, =4-2(-1)+1, =44+2+1, =7=RAHS.