Question

यदि `1,omega` तथा `omega^(2)` इकाई के घनमूल है तो सिद्ध कीजिए कि -
(i) `(xomega+yomega^(2))(xomega^(2)+yomega)=x^(2)-xy+y^(2)` (ii) `[(1-omega+omega^(2))(1+omega-omega^(2))]^(2)=16`
(iii) `(-1)^(1//3)=-1,-omega,-omega^(2)` (iv) `(-8)^(1//3)=-2,-2omega,-2omega^(2)`
(v) `(1+omega)(1+omega^(2))(1+omega^(4))(1+omega^(5))=1`

Answer 1

(i) बायाँ पक्ष `=(xomega+yomega^(2))(xomega^(2)+yomega)=xomega(xomega^(2)+yomega)+yomega^(2)(xomega^(2)+yomega)`
`=x^(2)omega^(3)+xyomega^(2)+xyomega^(4)+y^(2)omega^(3)`
`=(x^(2)+y^(2))omega^(3)+xyomega^(2)(1+omega^(2))=(x^(2)+y^(2))*1+xyomega^(2)(-omega)`
`=x^(2)+y^(2)-xyomega^(3)=x^(2)+y^(2)-xy=` दायाँ पक्ष