## Expanding (a + b)^9: A Journey with the Binomial Theorem

The expression (a + b)^9 might seem daunting at first glance, but with the help of the binomial theorem, we can break it down into a manageable and elegant expansion. Let's explore this powerful tool and its application.

### Understanding the Binomial Theorem

The binomial theorem provides a formula for expanding expressions of the form (x + y)^n, where n is a non-negative integer. It states:

**(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 choose k) = n! / (k! * (n-k)!)**

This formula might seem complex, but it's just a systematic way of expressing the pattern observed in binomial expansions.

### Expanding (a + b)^9

Let's apply the binomial theorem to expand (a + b)^9:

**(a + b)^9 = (9 choose 0) * a^9 * b^0 + (9 choose 1) * a^8 * b^1 + (9 choose 2) * a^7 * b^2 + ... + (9 choose 9) * a^0 * b^9**

Now, let's calculate the binomial coefficients:

**(9 choose 0) = 1****(9 choose 1) = 9****(9 choose 2) = 36****(9 choose 3) = 84****(9 choose 4) = 126****(9 choose 5) = 126****(9 choose 6) = 84****(9 choose 7) = 36****(9 choose 8) = 9****(9 choose 9) = 1**

Substituting these values back into the expanded expression:

**(a + b)^9 = 1 * a^9 * b^0 + 9 * a^8 * b^1 + 36 * a^7 * b^2 + 84 * a^6 * b^3 + 126 * a^5 * b^4 + 126 * a^4 * b^5 + 84 * a^3 * b^6 + 36 * a^2 * b^7 + 9 * a^1 * b^8 + 1 * a^0 * b^9**

Simplifying the expression:

**(a + b)^9 = a^9 + 9a^8b + 36a^7b^2 + 84a^6b^3 + 126a^5b^4 + 126a^4b^5 + 84a^3b^6 + 36a^2b^7 + 9ab^8 + b^9**

This is the complete expansion of (a + b)^9.

### Key Takeaways

The binomial theorem is a powerful tool that allows us to expand expressions of the form (x + y)^n in a systematic and efficient way. It simplifies the process of multiplying these expressions, revealing the underlying patterns in the expansion. This understanding is invaluable in various fields, including algebra, calculus, and probability.