Factoring and Simplifying the Expression: (a²b²)³ + (b²c²)³ + (c²a²)³ / (ab)³ + (bc)³ + (ca)³
This expression presents a challenge in simplification due to its complex structure. However, we can simplify it using the principles of factoring and algebraic manipulation.
Understanding the Key Concepts
1. Sum of Cubes Formula:
The sum of cubes formula states: a³ + b³ = (a + b)(a²  ab + b²)
2. Difference of Squares Formula:
The difference of squares formula states: a²  b² = (a + b)(a  b)
3. Factoring by Grouping:
Sometimes, we can factor expressions by grouping terms with common factors.
Simplifying the Expression
Let's break down the simplification process:

Factor the Numerator:
 Notice that the numerator consists of three terms, each representing the sum of cubes. We can apply the sum of cubes formula to each term.
(a²b²)³ + (b²c²)³ + (c²a²)³ = [(a²b²)+(b²c²)][(a²b²)²  (a²b²)(b²c²) + (b²c²)²] + (c²a²)³
 Further factor the difference of squares terms within the expression:
= [(a+b)(ab) + (b+c)(bc)][(a²b²)²  (a²b²)(b²c²) + (b²c²)²] + (c+a)(ca)[(c²a²)²  (c²a²)(c²b²) + (c²b²)²]

Factor the Denominator:
 The denominator also consists of three terms, representing the sum of cubes. Applying the sum of cubes formula:
(ab)³ + (bc)³ + (ca)³ = [(ab)+(bc)][(ab)²  (ab)(bc) + (bc)²] + (ca)³
 Further factor by grouping:
= [(ab)+(bc)][(ab)²  (ab)(bc) + (bc)²] + (ca)(ca)[(ca)² + (ca)(cb) + (cb)²]

Cancellation and Simplification:
 Notice that both the numerator and denominator share common factors: (ab), (bc), and (ca). Canceling these common factors:
= [(a+b)(b+c)][(a²b²)²  (a²b²)(b²c²) + (b²c²)²] + (c+a)[(c²a²)²  (c²a²)(c²b²) + (c²b²)²] / [(ab)+(bc)][(ab)²  (ab)(bc) + (bc)²] + (ca)[(ca)² + (ca)(cb) + (cb)²]

Final Result:
 The final simplified expression is:
= [(a+b)(b+c)][(a²b²)²  (a²b²)(b²c²) + (b²c²)²] + (c+a)[(c²a²)²  (c²a²)(c²b²) + (c²b²)²] / [(ab)+(bc)][(ab)²  (ab)(bc) + (bc)²] + (ca)[(ca)² + (ca)(cb) + (cb)²]
Conclusion
This expression can be simplified using algebraic manipulation and factoring techniques. The process involves applying the sum of cubes formula, the difference of squares formula, and factoring by grouping. The final simplified expression may still appear complex, but it's a significantly reduced form of the original expression.