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]#?
1 Answer
#((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
Explanation:
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