## Expanding (x + 2y)⁶ with the Binomial Theorem

The binomial theorem provides a powerful tool for expanding expressions of the form (x + y)ⁿ. Let's explore how it works when expanding (x + 2y)⁶.

### The Binomial Theorem

The binomial theorem states:

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

Where:

**(n choose k)**is the binomial coefficient, calculated as n!/(k!(n-k)!). It represents the number of ways to choose k objects from a set of n objects.**∑_(k=0)^n**represents the sum from k=0 to n.

### Expanding (x + 2y)⁶

Let's apply the binomial theorem to our expression:

**(x + 2y)⁶ = ∑_(k=0)^6 (6 choose k) x^(6-k) (2y)^k**

We need to calculate each term in the sum:

**k = 0:** (6 choose 0) x⁶ (2y)⁰ = 1 * x⁶ * 1 = **x⁶**

**k = 1:** (6 choose 1) x⁵ (2y)¹ = 6 * x⁵ * 2y = **12x⁵y**

**k = 2:** (6 choose 2) x⁴ (2y)² = 15 * x⁴ * 4y² = **60x⁴y²**

**k = 3:** (6 choose 3) x³ (2y)³ = 20 * x³ * 8y³ = **160x³y³**

**k = 4:** (6 choose 4) x² (2y)⁴ = 15 * x² * 16y⁴ = **240x²y⁴**

**k = 5:** (6 choose 5) x¹ (2y)⁵ = 6 * x¹ * 32y⁵ = **192xy⁵**

**k = 6:** (6 choose 6) x⁰ (2y)⁶ = 1 * 1 * 64y⁶ = **64y⁶**

Therefore, the complete expansion of (x + 2y)⁶ is:

**(x + 2y)⁶ = x⁶ + 12x⁵y + 60x⁴y² + 160x³y³ + 240x²y⁴ + 192xy⁵ + 64y⁶**

### Conclusion

The binomial theorem provides a systematic way to expand expressions of the form (x + y)ⁿ. By understanding the formula and applying it step-by-step, we can efficiently obtain the expanded form of any binomial expression.