## Expanding (a+b)^3

The expansion of (a+b)^3 is a fundamental concept in algebra, often used in various mathematical applications. Understanding its expansion is crucial for simplifying expressions, solving equations, and performing other algebraic operations.

### Understanding the Expansion

The expansion of (a+b)^3 represents the product of (a+b) multiplied by itself three times:

**(a+b)^3 = (a+b)(a+b)(a+b)**

To expand this, we can apply the distributive property multiple times:

**First expansion:**(a+b)(a+b) = a(a+b) + b(a+b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2**Second expansion:**(a^2 + 2ab + b^2)(a+b) = a(a^2 + 2ab + b^2) + b(a^2 + 2ab + b^2) = a^3 + 2a^2b + ab^2 + ba^2 + 2ab^2 + b^3 = a^3 + 3a^2b + 3ab^2 + b^3

Therefore, the complete expansion of (a+b)^3 is:

**(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3**

### Applying the Binomial Theorem

The expansion of (a+b)^3 can also be derived using the **Binomial Theorem**, which provides a general formula for expanding any binomial raised to a power. The binomial theorem states:

**(a+b)^n = ∑(n choose k) a^(n-k) b^k**

where:

**n**is the power to which the binomial is raised**k**is a non-negative integer ranging from 0 to n**(n choose k)**is the binomial coefficient, calculated as n! / (k! * (n-k)!)

For (a+b)^3, we have n = 3. Applying the binomial theorem:

**(a+b)^3 = (3 choose 0)a^3b^0 + (3 choose 1)a^2b^1 + (3 choose 2)a^1b^2 + (3 choose 3)a^0b^3**

Calculating the binomial coefficients:

- (3 choose 0) = 3! / (0! * 3!) = 1
- (3 choose 1) = 3! / (1! * 2!) = 3
- (3 choose 2) = 3! / (2! * 1!) = 3
- (3 choose 3) = 3! / (3! * 0!) = 1

Substituting the coefficients back into the expansion:

**(a+b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3 = a^3 + 3a^2b + 3ab^2 + b^3**

### Conclusion

Expanding (a+b)^3 results in the expression a^3 + 3a^2b + 3ab^2 + b^3. This expansion can be obtained through repeated application of the distributive property or by utilizing the Binomial Theorem. Understanding this expansion is essential for various algebraic manipulations and applications.