Question

if 1,w,w² are are the cube roots of unity (1-w+w²)⁵+(1+w-w²)⁵=32
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Answer 1

Answer:

we know 1+w+w²=0 , w³=1

here,

here,1+w² = -w and 1+w = -w² and w+w²=-1..

Apply in equation..

=> (1-w+w²)⁵ + (1+w-w²)⁵

=>(1+w²-w)⁵ + (1+w-w²)⁵

=>(-w-w)⁵ + (-w²-w²)⁵

=>(-2w)⁵ + (-2w²)⁵

=>-2⁵(w⁵ + w¹⁰)

=>-2⁵(w² + w)

=>-2⁵(-1)

=>-32(-1)

=>32...

Hence it is proved..

I HOPE THIS ANSWER HELPS YOU...