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Question
State whether the following statement is true or false.If ω is one of the complex cube roots of unity, then
(1−ω+ω2)3=(1+ω−ω2)3=−8
State whether the following statement is true or false.
(1−ω+ω2)3=(1+ω−ω2)3=−8
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Solution
The correct option is A True
We know that if w is complex cube root of unity then (1+w+w2)=0 and w3=1..........[1]
So, (1+w2)=−w and (1+w=−w2)..........[from 1].........[2]
Now,
(1+w2−w)3=(−w−w)3=(−2w)3=(−2)3(w)3=−8(1)=−8
Also,
(1−w2+w)3=(−w2−w2)3=(−2w2)3=(−2)3(w2)3=−8(w6)=−8
So,
(1−w+w2)3=(1+w−w2)3=−8
We know that if w is complex cube root of unity then (1+w+w2)=0 and w3=1..........[1]
So, (1+w2)=−w and (1+w=−w2)..........[from 1].........[2]
Now,
(1+w2−w)3=(−w−w)3=(−2w)3=(−2)3(w)3=−8(1)=−8
Also,
(1−w2+w)3=(−w2−w2)3=(−2w2)3=(−2)3(w2)3=−8(w6)=−8
So,
(1−w+w2)3=(1+w−w2)3=−8
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