How do you simplify ## [ (a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3 ] / [ (a^4 - b^4)^3 + (b^4 - c^4)^3 + (c^4 - a^4)^3]##?

##((a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3)/((a^4-b^4)^3+(b^4-c^4)^3+(c^4-a^4)^3)=1/((a^2+b^2)(b^2+c^2)(c^2+a^2))##

excluding any of ##a=+-b##, ##b=+-c##, ##c=+-a##.

Notice that:

##(A-B)^3+(B-C)^3+(C-A)^3##

##=(color(red)(cancel(color(black)(A^3)))-3A^2B+3AB^2-color(red)(cancel(color(black)(B^3))))+(color(red)(cancel(color(black)(B^3)))-3B^2C+3BC^2-color(red)(cancel(color(black)(C^3))))+(color(red)(cancel(color(black)(C^3)))-3C^2A+3CA^2-color(red)(cancel(color(black)(A^3))))##

##=3(AB^2-A^2B+BC^2-B^2C+CA^2-C^2A)##

##=3(A-B)(B-C)(C-A)##

Note also:

##a^4-b^4 = (a^2-b^2)(a^2+b^2)##, etc.

So:

##((a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3)/((a^4-b^4)^3+(b^4-c^4)^3+(c^4-a^4)^3)##

##=(color(red)(cancel(color(black)(3)))(a^2-b^2)(b^2-c^2)(c^2-a^2))/(color(red)(cancel(color(black)(3)))(a^4-b^4)(b^4-c^4)(c^4-a^4))##

##=(color(red)(cancel(color(black)((a^2-b^2))))color(red)(cancel(color(black)((b^2-c^2))))color(red)(cancel(color(black)((c^2-a^2)))))/(color(red)(cancel(color(black)((a^2-b^2))))(a^2+b^2)color(red)(cancel(color(black)((b^2-c^2))))(b^2+c^2)color(red)(cancel(color(black)((c^2-a^2))))(c^2+a^2))##

##=1/((a^2+b^2)(b^2+c^2)(c^2+a^2))##

excluding any of ##a=+-b##, ##b=+-c##, ##c=+-a##