If `(x^(3)+y^(3) )prop (x^(3)-y^(3)) `, then prove that (x+y) `prop` (x-y).

Explanation

`(x^(3)+y^(3)) prop(x^(3)-y^(3))`
` rArr x^(3) +y^(3)=k(x^(3) -y^(3))[when k ne0="variation constant"]`
`rArr (x^(3)+y^(3))/(x^(3)-y^(3))=krArr(x^(3)+y^(3)+x^(3)-y^(3))/(x^(3)+y^(3)-x^(3)+y^(3))=(k+1)/(k-1)` [by componendo and dividendo ]
`rArr (2x^(3))/(2y^(3))=(k+1)/(k-1)rArr (x^(3))/(y^(3))=(k+1)/(k-1)rArr (x/y)^(3) =(k+1)/(k-1)`
`rArrx/y =3sqrt((k+1)/(k-1))=m("let")ne0`.
`rArr(x+y)/(x-y)=(m+1)/(m-1)`(by componendo and dividendo )
`rArr(x+y)/(x-y)=n [when n=(m+1)/(m-1)ne0]`
`rArr x+y =n (x-y) `
`therefore (x+y) prop(x-y) (proved )[because n ne0 =` variation constant ]