To expand the expression (x + y)^8

To expand the expression (x + y)^8, we use the binomial theorem, which states that (x + y)^n = Σ (n choose k) * x^(n-k) * y^k, where the sum is from k = 0 to n.

Here, n = 8. So, we calculate each term of the sum:

1. For k = 0: (8 choose 0) * x^8 * y^0 = 1 * x^8 * 1 = x^8
2. For k = 1: (8 choose 1) * x^7 * y^1 = 8 * x^7 * y = 8x^7y
3. For k = 2: (8 choose 2) * x^6 * y^2 = 28 * x^6 * y^2 = 28x^6y^2
4. For k = 3: (8 choose 3) * x^5 * y^3 = 56 * x^5 * y^3 = 56x^5y^3
5. For k = 4: (8 choose 4) * x^4 * y^4 = 70 * x^4 * y^4 = 70x^4y^4
6. For k = 5: (8 choose 5) * x^3 * y^5 = 56 * x^3 * y^5 = 56x^3y^5
7. For k = 6: (8 choose 6) * x^2 * y^6 = 28 * x^2 * y^6 = 28x^2y^6
8. For k = 7: (8 choose 7) * x^1 * y^7 = 8 * x * y^7 = 8xy^7
9. For k = 8: (8 choose 8) * x^0 * y^8 = 1 * 1 * y^8 = y^8

So, (x + y)^8 = x^8 + 8x^7y + 28x^6y^2 + 56x^5y^3 + 70x^4y^4 + 56x^3y^5 + 28x^2y^6 + 8xy^7 + y^8.