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$a_{3}+b_{3 formula}$ 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:

${\mathrm{a3+b3}}^{}=(a+b)({a}^{2}-\mathrm{ab}+{b}^{2})$

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

## Proving The a3+b3 Formula

The proof of $a_{3}+b_{3}$ 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, $(a+b)({a}^{2}-\mathrm{ab}+{b}^{2})$. This involves applying the distributive property of multiplication over addition.

$(a+b)({a}^{2}-\mathrm{ab}+{b}^{2})=a({a}^{2}-\mathrm{ab}+{b}^{2})+b({a}^{2}-\mathrm{ab}+{b}^{2})$

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.

$=a{}^{3}-{a}^{2}b+a{b}^{2}+b{a}^{2}-b\mathrm{ab}+{b}^{3}$

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.

$=a{}^{3}-{a}^{2}b+a{b}^{2}+{a}^{2}b-a{b}^{2}+{b}^{3}$

Notice how $-a{}^{2}b$ and $+{a}^{2}b$ cancel each other, as do $a{b}^{2}$ and $-a{b}^{2}$. This cancellation is a pivotal part of the proof.

**Step 4: Final Simplification**

After the cancellation, the equation simplifies to:

$=a{}^{3}+{b}^{3}$

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

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## Conclusion

Through systematic expansion, distribution, and simplification of a3+b3, we have shown that a3+b3${}^{}=(a+b)({a}^{2}-\mathrm{ab}+{b}^{2})$. 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.