We are given the following expression to simplify:

$\frac{(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3}{(a - b)^3 + (b - c)^3 + (c - a)^3}$

Step 1: Examine the structure of the expression

We can recognize that both the numerator and the denominator contain a similar structure involving cubes of differences. Let's first recall that:

$a^2 - b^2 = (a - b)(a + b)$

This will allow us to simplify each term in the numerator.

Step 2: Simplifying the terms

• Each term in the numerator is of the form $(a^2 - b^2)^3$, which can be written as: $(a^2 - b^2)^3 = [(a - b)(a + b)]^3 = (a - b)^3(a + b)^3$

  Applying the same for the other terms: $(b^2 - c^2)^3 = (b - c)^3(b + c)^3$ $(c^2 - a^2)^3 = (c - a)^3(c + a)^3$

Thus, the entire numerator becomes: $(a - b)^3(a + b)^3 + (b - c)^3(b + c)^3 + (c - a)^3(c + a)^3$

Step 3: Analyze the denominator

The denominator is already in a simplified form: $(a - b)^3 + (b - c)^3 + (c - a)^3$

Step 4: Factor the expression

Notice that both the numerator and denominator share common terms of $(a - b)^3$, $(b - c)^3$, and $(c - a)^3$. Hence, we can factor out the common $(a - b)^3$, $(b - c)^3$, and $(c - a)^3$ from the numerator and denominator.

This simplifies the expression to: [ \frac{(a - b)^3[(a + b)^3] + (b - c)^3[(b + c)^3] + (c - a)^3[(c + a)^3]}{(a - b)^3 + (b - c)^3 + (c - a)^3} ]

Step 5: Conclusion

The expression simplifies further as each factor in the numerator and denominator cancels out. Therefore, the entire expression simplifies to:

$(a + b) + (b + c) + (c + a)$

Finally, simplifying this sum gives:

$\boxed{0}$

The entire expression simplifies to zero.

Would you like further clarification on any of these steps?

Related Questions:

1. How do you simplify algebraic expressions involving powers and sums?
2. What is the general process for factoring out common terms in expressions?
3. How do you recognize when an expression can be factored using difference of squares?
4. How does symmetry in expressions help in simplifying algebraic terms?
5. Can you apply this process to other similar cubic expressions?

Tip:

Always look for common factors or patterns in both the numerator and denominator when simplifying complex rational expressions.