Question no. 6

Question no. 6 e of 3 then 0 and are the complex cube roots Of unity. show that S) If and 13 are the complex cube roots of unity, show that -113-1 = o. 6) If x=a+b, y=åa-+ßb and z=aß+bu where and ß are complex cube roots of unity, show that .0z=a3+b3- 7) If w is the complex cube-root of unity, show that 8) If w is the complex cube-root of unity, show that _ 9) If w denotes the complex cube root of unity, prove the following Im vv is the comolex cube root of unity then prove

Dear student

W e h a v e, x = a + b, y = α a + β b, z = α b + β a N o w, a s α, β a r e c o m p l e x c u b e r o o t o f u n i t y s o w e h a v e, α β = 1, 1 + α + β = 0, α^{2} = β, β^{2} = α s o x y z = (a + b) (α a + β b) (α b + β a) = (a + b) (a b α^{2} + a^{2} α β + b^{2} α β + a b β^{2}) = (a + b) (a^{2} + b^{2} + a b (α^{2} + β^{2})) = (a + b) (a^{2} + b^{2} + a b (α + β)) = (a + b) (a^{2} + b^{2} + a b (- 1)) = (a + b) (a^{2} + b^{2} - a b) = a^{3} + b^{3}

Hope this clears your doubt
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