To expand the expression (x + y)^8 using Pascal's triangle and the binomial theorem

To expand the expression (x + y)^8 using Pascal's triangle and the binomial theorem, follow these steps:

1. Write down the 8th row of Pascal's triangle. Pascal's triangle is a triangular array of binomial coefficients. The 8th row (starting from 0) is: 1, 8, 28, 56, 70, 56, 28, 8, 1.

2. Write down the binomial expression (x + y)^8.

3. The binomial theorem states that (x + y)^n = Σ (n choose k) * x^(n-k) * y^k, where the sum is from k = 0 to n. In this case, n = 8.

4. Expand the expression using the coefficients from Pascal's triangle and the powers of x and y from the binomial theorem:

   • For k = 0: 1 * x^8 * y^0 = x^8
   • For k = 1: 8 * x^7 * y^1 = 8x^7y
   • For k = 2: 28 * x^6 * y^2 = 28x^6y^2
   • For k = 3: 56 * x^5 * y^3 = 56x^5y^3
   • For k = 4: 70 * x^4 * y^4 = 70x^4y^4
   • For k = 5: 56 * x^3 * y^5 = 56x^3y^5
   • For k = 6: 28 * x^2 * y^6 = 28x^2y^6
   • For k = 7: 8 * x^1 * y^7 = 8xy^7
   • For k = 8: 1 * x^0 * y^8 = y^8

5. Combine all the terms to get the expanded expression: (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.