Exploring the Algebraic Identity: (ab)^3 + (bc)^3 + (ca)^3 / 9(ab)(bc)(ca)
This article delves into the interesting algebraic identity:
(ab)^3 + (bc)^3 + (ca)^3 / 9(ab)(bc)(ca)
We will explore its derivation, significance, and applications.
Derivation of the Identity
The identity can be derived using the following steps:

Factoring the numerator: We can factor the numerator using the sum of cubes formula:
(ab)^3 + (bc)^3 + (ca)^3 = 3(ab)(bc)(ca)

Simplifying the expression: Substituting the factored numerator in the original expression:
3(ab)(bc)(ca) / 9(ab)(bc)(ca) = 1/3
Therefore, the identity simplifies to 1/3.
Significance of the Identity
This identity is significant because it reveals a relationship between the cubes of differences and the product of those differences. It provides a concise and elegant way to express this relationship.
Applications
The identity can be applied in various mathematical contexts, including:
 Algebraic manipulation: It can simplify complex expressions involving cubes of differences.
 Proofs: The identity can be used to prove other mathematical theorems or identities.
 Problem solving: It can help in solving problems related to algebraic expressions involving cubes of differences.
Example
Let's consider a simple example:
If a = 2, b = 1, and c = 0, then:
(21)^3 + (10)^3 + (02)^3 / 9(21)(10)(02) = 1^3 + 1^3 + (2)^3 / 9(1)(1)(2) = 1 + 1  8 / 18 = 6/18 = 1/3.
This confirms that the identity holds true for these values.
Conclusion
The algebraic identity (ab)^3 + (bc)^3 + (ca)^3 / 9(ab)(bc)(ca) = 1/3 is a valuable tool in algebra. It simplifies expressions, provides elegant relationships, and can be applied in diverse mathematical situations.