Question

(a²-b²)³+ (b²-c²)³+(c²-a²)³/(a-b)³+(b-c)³+(c-a)³​

Answer 1

hope..it helps..

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Answer 2

$\bold{ \red{ \underline{ \underline{ \huge{ \mid{ \: \: SOLUTION \: \: { \mid}}}}}}}$

$\bold{We \: \: know \: \: that , } \\ \\ \boxed{\bold{If \: \: x + y + z = 0 \: \: then \: \: x {}^{3} + y {}^{3} + z {}^{3} = 3xyz}}$

$\bold{We \: \: can \: \: observe \: \: that,} \\ \\ \bold{(a {}^{2} - b {}^{2}) + ( b {}^{2} - c {}^{2}) + ( c {}^{2} - a {}^{2}) = 0 } \\ \\ \bold{ Then,} \\ \\ \underline{\bold{(a {}^{2} - b {}^{2}) {}^{3} + ( b {}^{2} - c {}^{2}) {}^{3} + ( c {}^{2} - a {}^{2}) {}^{3} =3(a {}^{2} - b {}^{2}) ( b {}^{2} - c {}^{2}) ( c {}^{2} - a {}^{2}) }}$

$\bold{Similarly ,\: \: we \: \: can \: \: observe \: \: : } \\ \\ \bold{(a - b) + (b - c) + (c - a) = 0} \\ \\ \bold{Then} \\ \\ \underline{\bold{(a - b) {}^{3} + (b - c) {}^{3} (c - a) {}^{3} = 3( a - b)(b - c)(c - a)}}$

$\bold{Now,}\\ \\ \bold{ \frac{(a {}^{2} - b {}^{2}) {}^{3} + (b {}^{2} - c {}^{2}) {}^{3} + (c {}^{2} - a {}^{2} ) {}^{3} }{(a - b) {}^{3} + (b - c) {}^{3} + ( c- a) {}^{3} } } \\ \\ \bold{ = \frac{3(a {}^{2} - b {}^{2}) ( b {}^{2} - c {}^{2}) ( c {}^{2} - a {}^{2})}{3( a - b)(b - c)(c - a)} } \\ \\ \bold{ = \frac{(a {}^{2} - b {}^{2}) ( b {}^{2} - c {}^{2}) ( c {}^{2} - a {}^{2})}{( a - b)(b - c)(c - a)} } \\ \\ \bold{ = \frac{(a + b)( a - b)( b+ c)(b - c)( c+a )(c - a)}{( a - b)(b - c)(c - a)} } \\ \\ \bold{ = ( a + b)(b + c)(c + a)}$

$\bold{ \red{ \underline{ \underline{ \huge{ \mid{ \: \: ANSWER\: \: { \mid}}}}}}}$➡


$\boxed{ \boxed{ \purple{\bold{ \frac{(a {}^{2} - b {}^{2}) {}^{3} + (b {}^{2} - c {}^{2}) {}^{3} + (c {}^{2} - a {}^{2} ) {}^{3} }{(a - b) {}^{3} + (b - c) {}^{3} + ( c- a) {}^{3} } \: = (a +b )( b+ c)( c+ a)}}}}$

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✝$\pink{ \mathcal{ \: THANKS.. \: }}$✝