## Exploring the Expansion of (a + bi)³

The expression (a + bi)³ represents the cube of a complex number, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (√-1). Expanding this expression can be done using the binomial theorem or by direct multiplication.

### Understanding the Binomial Theorem

The binomial theorem provides a general formula for expanding expressions of the form (x + y)ⁿ:

(x + y)ⁿ = ∑_(k=0)^n (n choose k) x^(n-k) y^k

Where (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).

Applying this to (a + bi)³, we get:

(a + bi)³ = (3 choose 0) a³ (bi)⁰ + (3 choose 1) a² (bi)¹ + (3 choose 2) a¹ (bi)² + (3 choose 3) a⁰ (bi)³

Simplifying this expression:

(a + bi)³ = a³ + 3a²bi - 3ab² - b³i

### Direct Multiplication Approach

Alternatively, we can expand (a + bi)³ by directly multiplying the expression three times:

(a + bi)³ = (a + bi)(a + bi)(a + bi)

Expanding the first two factors:

(a + bi)(a + bi) = a² + 2abi - b²

Multiplying this result with (a + bi):

(a² + 2abi - b²)(a + bi) = a³ + 3a²bi - 3ab² - b³i

Both approaches lead to the same result:

**(a + bi)³ = a³ + 3a²bi - 3ab² - b³i**

### Conclusion

The expansion of (a + bi)³ reveals the complex nature of the result. It is a complex number with both real and imaginary components. Understanding this expansion is crucial for various applications in mathematics, particularly in algebra, trigonometry, and complex analysis.