Question

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

Answer 1

Answer:

On simplifying we get ( a + b )( b + c )( c + a )

Step-by-step explanation:

Given Expression:

We need to simplify given expression.

We use the following result,

if x + y + z = 0 then x³ + y³ + z³ = 3xyz

First x = a² - b² , y = b² - a² and z = c² - a²

⇒ x + y + z = a² - b² + b² - c² + c² - a² = 0

So, (a² - b²)³ + (b² - a²)³ + (c² - a²) = 3(a² - b²)(b² - a²)(c² - a²)

Second, x = a - b  ,  y = b - c  and z = c - a

⇒ x + y + z = a - b + b - c + c - a = 0

So,  (a - b)³ + (b - c)³ + (c - a)³ = 3(a - b)(b - c)(c - a)

Thus, we get (write answer on own)

Now using, x² - y² = ( x - y )( x + y )

Therefore, On simplifying we get ( a + b )( b + c )( c + a )

Answer 2

$\huge{ \boxed{ \rm{ \red{answer}}}}$

$\green{ \frac{ {({a}^{2} - {b}^{2} )}^{3} + { {(b}^{2} - {c}^{2}) }^{3} + { ({c}^{2} - {a}^{2}) }^{3} }{ {(a - b)}^{3} + {(b - c)}^{3} + {(c - a)}^{3} } } \\ \\ = \orange{a - b + b - c + c - a} \\ \\ = \: a - a \\ \\ = \red{0}$

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