Simplify:((a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3)/((a-b)^3+(b-c)^3+(c-a)^3)

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We have to simplify: ((a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3)/((a-b)^3+(b-c)^3+(c-a)^3)

(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3 = 3(b-a)(b+a)(c-a)(c+a)(c-b)(c+b)...(1)

(a-b)^3+(b-c)^3+(c-a)^3 = 3(b-a)(c-a)(c-b)...(2)

Dividing (1)/(2), we get

((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(b-a)(b+a)(c-a)(c+a)(c-b)(c+b)/3(b-a)(c-a)(c-b)

=> (b+a)(c+a)(c+b)

The simplified form of ((a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3)/((a-b)^3+(b-c)^3+(c-a)^3) = (b+a)(c+a)(c+b)

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We have to simplify: ((a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3)/((a-b)^3+(b-c)^3+(c-a)^3)

(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3 = 3(b-a)(b+a)(c-a)(c+a)(c-b)(c+b)...(1)

(a-b)^3+(b-c)^3+(c-a)^3 = 3(b-a)(c-a)(c-b)...(2)

Dividing (1)/(2), we get

((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(b-a)(b+a)(c-a)(c+a)(c-b)(c+b)/3(b-a)(c-a)(c-b)

=> (b+a)(c+a)(c+b)

The simplified form of ((a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3)/((a-b)^3+(b-c)^3+(c-a)^3) = (b+a)(c+a)(c+b)

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