a3+b3 formula

Prove: a3+b3 Formula


 is a cornerstone in algebra, serving as a critical tool for simplifying and solving cubic expressions. Its relevance spans various mathematical disciplines, from basic algebra to advanced calculus. Understanding this formula not only enhances problem-solving skills but also lays the groundwork for tackling more complex algebraic challenges. The formula is expressed as:


This article delves into a detailed proof of this formula, illuminating its derivation and practical significance.

Proving The a3+b3 Formula

a3+b3 formula

The proof of hinges on demonstrating that the expanded and simplified form of the right-hand side (RHS) of the equation is equal to the left-hand side (LHS).

Step 1: Expand the Right-Hand Side

First, we focus on expanding the RHS of the equation, (a +b)(a2ab+b2). This involves applying the distributive property of multiplication over addition.

(a +b)(a2ab+b2)=a(a2ab+b2)+b(a2ab+b2)

This expansion is the first critical step in unraveling the formula.

Step 2: Distribute the Terms

Next, we distribute the terms and within their respective brackets. This step is crucial as it lays the foundation for the subsequent simplification.

=a 3a2b+ab2+ba2bab+b3

Each term is carefully multiplied, ensuring that all possible combinations are accounted for.

Step 3: Simplify the Expression

In this step, we simplify by combining like terms. It’s important to identify and cancel out terms that negate each other.

=a 3a2b+ab2+a2bab2+b3

Notice how −a 2b and +a2b cancel each other, as do ab2 and ab2. This cancellation is a pivotal part of the proof.

Step 4: Final Simplification

After the cancellation, the equation simplifies to:

=a 3+b3

This final step confirms that the RHS of our original formula simplifies to match the LHS exactly.

Must Read:


Through systematic expansion, distribution, and simplification of a3+b3, we have shown that a3+b3 =(a+b)(a2ab+b2). This proof not only validates the formula but also highlights the elegance of algebraic manipulation. The ability to factorize and simplify cubic expressions using this formula is a valuable skill in mathematics.

Leave a Reply

Your email address will not be published. Required fields are marked *