The Curious Case of (ab)³ + (bc)³ + (ca)³
The expression (ab)³ + (bc)³ + (ca)³ might appear intimidating at first glance, but it holds a fascinating mathematical secret. This article will explore the simplification of this expression and unveil its remarkable property.
Understanding the Algebraic Identity
The key to simplifying this expression lies in recognizing a fundamental algebraic identity:
x³ + y³ + z³  3xyz = (x + y + z)(x² + y² + z²  xy  xz  yz)
This identity provides a powerful tool for manipulating expressions involving cubes. To utilize this identity, we need to cleverly manipulate our expression.
Applying the Identity

Substitution: Let's make the following substitutions:
 x = (ab)
 y = (bc)
 z = (ca)

Expansion: Notice that (x + y + z) = (ab) + (bc) + (ca) = 0.

Simplification: Using the identity, we have: (ab)³ + (bc)³ + (ca)³ = 3(ab)(bc)(ca)
The Surprising Result
The final result reveals that the seemingly complex expression (ab)³ + (bc)³ + (ca)³ simplifies to 3(ab)(bc)(ca). This surprising outcome demonstrates the power of algebraic identities to condense and simplify complex mathematical expressions.
Conclusion
The simplification of (ab)³ + (bc)³ + (ca)³ showcases the elegance and utility of algebraic identities. Understanding these identities empowers us to manipulate and simplify complex expressions, making them more manageable and revealing hidden relationships within the world of mathematics.