यदि ` omega ` इकाई का काल्पनिक घनमूल हो तो साबित कीजिये कि ` ( 1 + omega ) ( 1 + omega ^ 2 ) ( 1 + omega ^ 4 ) (1 + omega ^ 8 ) ... ` to n factors = ` 1 or - omega ^ 2 ` जबकि n क्रमशः सम और विषम है

यदि ` omega ` इकाई का काल्पनिक घ�

1.	यदि ` omega ` इकाई का काल्पनिक घनमूल हो तो साबित कीजिये कि ` ( 1 + omega ) ( 1 + omega ^ 2 ) ( 1 + omega ^ 4 ) (1 + omega ^ 8 ) ... ` to n factors = ` 1 or - omega ^ 2 ` जबकि n क्रमशः सम और विषम है
Answer» LHS = ` ( 1 + omega ) ( 1 + omega ^ 2 ) ( 1 + omega ^ 4 ) ( 1 + omega ^8 ) ... ` to n factors ` = ( 1 + omega ) ( 1 + omega ^ 2 ) ( 1 + omega ) ( 1 + omega ^ 2) ...` to n factors ` = ( - omega ^ 2 ) ( - omega ) ( - omega ^ 2 ) ( - omega ) ... ` to n factors `" " `..(i) Case I. When n = 2m (even ) LHS = ` ( - omega ^ 2 ) ( - omega ) ( - omega ^ 2 ) ( - omega ) ... ` to 2m factors ` = [ ( - omega ^ 2 ) ( - omega ) ][ ( - omega ) ^ 2 ( - omega ) ]... ` to m factors ` = omega ^ 3 . omega ^ 3 . omega ^ 3... ` to m factors = 1 Case III. When n = 2m + 1 (odd ) LHS = ` ( - omega ^ 2 ) ( - omega ) ( - omega ^ 2 ) ( - omega ) ( - omega ^ 2 ) ... ` to ( 2m + 1 ) factors ` = ( - omega ^ 2 ) [ ( - omega ) ( - omega^ 2 ) ( - omega ) ( - omega ^ 2 ) ... ` to 2m factors ] ` = ( - omega ^ 2 ) [{ ( - omega ) ( - omega ^ 2 ) }{ ( - omega ) ( - omega ^ 2 ) }... ` to m factors ] ` = { ( - omega ^ 2 ) ( omega ^ 3 . omega ^ 3 . omega ^ 3 ) }... ` to m factors ) ` = ( - omega ^ 2) ( 1.1.1... ` to m factors ) ` = - omega ^ 2 `