(a^2-b^2)^3+(b^2-c^2)^3+(c^2-4^2)^3/(a-b)³+(b-c)³+(c-a)^3
Answers
Answer:
Answer:
On simplifying we get ( a + b )( b + c )( c + a )
Step-by-step explanation:
Given Expression:
\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)
3
+(b−c)
3
+(c−a)
3
(a
2
−b
2
)
3
+(b
2
−c
2
)
3
+(c
2
−a
2
)
3
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
\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)
3
+(b−c)
3
+(c−a)
3
(a
2
−b
2
)
3
+(b
2
−c
2
)
3
+(c
2
−a
2
)
3
=\frac{3(a^2-b^2)(b^2-c^2)(c^2-a^2)}{3(a-b)(b-c)(c-a)}=
3(a−b)(b−c)(c−a)
3(a
2
−b
2
)(b
2
−c
2
)(c
2
−a
2
)
Now using, x² - y² = ( x - y )( x + y )
=\frac{3(a-b)(a+b)(b-c)(b+c)(c-a)(c+a)}{3(a-b)(b-c)(c-a)}=
3(a−b)(b−c)(c−a)
3(a−b)(a+b)(b−c)(b+c)(c−a)(c+a)
(a+b)(b+c)(c+a)(a+b)(b+c)(c+a)