## Exploring the Power of Complex Numbers: (a + bi)^4

In the realm of mathematics, complex numbers are a fascinating and powerful tool. One intriguing aspect is understanding how to raise a complex number to a higher power. Let's delve into the expansion of (a + bi)^4, where 'a' and 'b' are real numbers and 'i' represents the imaginary unit (โ-1).

### The Binomial Theorem: Our Key to Expansion

To tackle this expansion, we turn to the trusty **Binomial Theorem**. This theorem states that for any real numbers 'x' and 'y' and a non-negative integer 'n':

**(x + y)^n = โ(n choose k) * x^(n-k) * y^k**

where the summation runs from k = 0 to n, and (n choose k) represents the binomial coefficient, calculated as n! / (k! * (n-k)!).

### Applying the Theorem to (a + bi)^4

Let's apply the binomial theorem to our complex number:

**(a + bi)^4 = โ(4 choose k) * a^(4-k) * (bi)^k**

Expanding this summation, we get:

**(a + bi)^4 = (4 choose 0) * a^4 * (bi)^0 + (4 choose 1) * a^3 * (bi)^1 + (4 choose 2) * a^2 * (bi)^2 + (4 choose 3) * a^1 * (bi)^3 + (4 choose 4) * a^0 * (bi)^4**

Now, we need to simplify each term:

**(4 choose 0) * a^4 * (bi)^0 = a^4****(4 choose 1) * a^3 * (bi)^1 = 4a^3bi****(4 choose 2) * a^2 * (bi)^2 = 6a^2b^2i^2 = -6a^2b^2**(Since i^2 = -1)**(4 choose 3) * a^1 * (bi)^3 = 4ab^3i^3 = -4ab^3i**(Since i^3 = -i)**(4 choose 4) * a^0 * (bi)^4 = b^4i^4 = b^4**(Since i^4 = 1)

### The Final Result

Combining all the terms, we arrive at the final expanded form of (a + bi)^4:

**(a + bi)^4 = a^4 + 4a^3bi - 6a^2b^2 - 4ab^3i + b^4**

This result shows that the fourth power of a complex number can be expressed as a combination of real and imaginary terms, effectively separating the real and imaginary components.

### Key Takeaways

- The binomial theorem provides a systematic approach to expanding powers of complex numbers.
- Understanding the properties of the imaginary unit 'i' (i^2 = -1, i^3 = -i, i^4 = 1) is crucial for simplifying the expansion.
- The expansion of (a + bi)^4 demonstrates the intriguing relationship between real and imaginary components in complex numbers.

By understanding the expansion of (a + bi)^4, we gain deeper insights into the power and versatility of complex numbers, which have far-reaching applications in various fields such as physics, engineering, and signal processing.