## Deriving the Cube of a Binomial: (a + b)³

The expansion of (a + b)³ is a fundamental concept in algebra, frequently used in various mathematical disciplines. Here's a step-by-step derivation of this formula:

### 1. Expanding the Expression

We can write (a + b)³ as the product of three identical binomials:

**(a + b)³ = (a + b)(a + b)(a + b)**

### 2. Applying the Distributive Property

Let's start by expanding the first two binomials:

**(a + b)(a + b) = a(a + b) + b(a + b)**

Using the distributive property, we get:

**(a + b)(a + b) = a² + ab + ba + b²**

Since multiplication is commutative, ab = ba, so we can simplify this to:

**(a + b)(a + b) = a² + 2ab + b²**

### 3. Expanding the Entire Expression

Now we need to multiply this result by the remaining (a + b):

**(a + b)³ = (a² + 2ab + b²)(a + b)**

Applying the distributive property once more:

**(a + b)³ = a²(a + b) + 2ab(a + b) + b²(a + b)**

Expanding each term:

**(a + b)³ = a³ + a²b + 2a²b + 2ab² + b²a + b³**

### 4. Combining Like Terms

Finally, we combine the terms with the same variables and exponents:

**(a + b)³ = a³ + 3a²b + 3ab² + b³**

### Conclusion

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

**(a + b)³ = a³ + 3a²b + 3ab² + b³**

This formula is useful for expanding expressions, simplifying equations, and understanding other algebraic concepts.