1.

सरल कीजिए । `((a^(2)-b^(2))^(3)+(b^(2)-c^(2))^(3)+(c^(2)-a^(2))^(3))/((a-b)^(3)+(b-c)^(3)+(c-a)^(3))`

Answer» यहाँ, `((a^(2)-b^(2))^(3)+(b^(2)-c^(2))^(3)+(c^(2)-a^(2))^(3))/((a-b)^(3)+(b-c)^(3)+(c-a)^(3))" "...(1)`
अब, `(a^(2)-b^(2))+(b^(2)-c^(2))+(c^(2)-a^(2))=0`
`therefore" "(a^(2)-b^(2))^(3)+(b^(2)-c^(2))^(3)+(c^(2)-a^(2))^(3)`
`= 3(a^(2)-b^(2))(b^(2)-c^(2))(c^(2)-a^(2))`
`= 3(a-b)(a+b)(b-c)(b+c)(c-a)(c+a)" "...(2)`
तथा, `(a-b)+(b-c)+(c-a)=0`
`therefore" "(a-b)^(3)+(b-c)^(3)+(c-a)^(3)=3(a-b)(b-c)(c-a)" "...(3)`
समीकरण (2) व (3) के मान समीकरण (1) में रखने पर
`((a^(2)-b^(2))^(3)+(b^(2)-c^(2))^(3)+(c^(2)-a^(2))^(3))/((a-b)^(3)+(b-c)^(3)+(c-a)^(3))=(3(a-b)(a+b)(b-c)(b+c)(c-a)(c+a))/(3(a-b)(b-c)(c-a))`
`= (a+b)(b+c)(c+a)`


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