यदि `omega ` इकाई का काल्पनिक (अवास्तविक) घनमूल हो तो सिद्ध कीजिये कि
` ( 1 - omega + omega ^ 2 ) ^ 5 + ( 1 + omega - omega ^ 2 ) ^ 5 = 32 `

Explanation

LHS ` = ( 1 - omega + omega ^ 2 ) ^ 5 + ( 1 + omega - omega ^ 2 ) ^ 5 `
` = ( 1 + omega ^ 2 - omega ) ^ 5 + ( 1 + omega - omega ^ 2 ) ^ 5 `
` = ( - omega - omega ) ^ 5 + ( - omega ^ 2 - omega ^ 2 ) ^ 5 [ because 1 + omega = - omega ^ 2 ` और ` 1 + omega ^ 2 = - omega ] `
` = ( - 2 omega ) ^ 5 + ( - 2 omega ^2 ) ^ 5 + ( - 2 omega ^ 2 ) ^ 5 = - 32 omega ^5 - 32 omega ^(10) = - 32omega^ 2 - 32 ( omega ^ 3 ) ^3. omega `
` = -32 omega ^ 2 - 32 omega = - 32 ( omega ^ 2 + omega ) = ( -32 ) xx ( -1 ) = 32 = ` RHS.