Find the value of (a + b)^4 when a = 2 and b = 3.

Find the value of (a + b)^4 when a = 2 and b = 3.

Step 1: Identify the values of a and b. Here, a = 2 and b = 3.
Step 2: Write down the expression (a + b)^4.
Step 3: Use the binomial theorem formula: (a + b)^n = C(n, 0)a^n b^0 + C(n, 1)a^(n-1)b^1 + ... + C(n, n)a^0b^n.
Step 4: For n = 4, the formula becomes: (a + b)^4 = C(4, 0)a^4b^0 + C(4, 1)a^3b^1 + C(4, 2)a^2b^2 + C(4, 3)a^1b^3 + C(4, 4)a^0b^4.
Step 5: Calculate the coefficients C(4, k) for k = 0, 1, 2, 3, 4: C(4, 0) = 1, C(4, 1) = 4, C(4, 2) = 6, C(4, 3) = 4, C(4, 4) = 1.
Step 6: Substitute a = 2 and b = 3 into the formula: (2 + 3)^4 = 1*(2^4)*(3^0) + 4*(2^3)*(3^1) + 6*(2^2)*(3^2) + 4*(2^1)*(3^3) + 1*(2^0)*(3^4).
Step 7: Calculate each term: 1*(16)*(1) + 4*(8)*(3) + 6*(4)*(9) + 4*(2)*(27) + 1*(1)*(81).
Step 8: Simplify each term: 16 + 96 + 216 + 216 + 81.
Step 9: Add all the terms together: 16 + 96 + 216 + 216 + 81 = 625.
Step 10: The final value of (a + b)^4 when a = 2 and b = 3 is 625.

Binomial Theorem – The binomial theorem provides a formula for expanding expressions of the form (a + b)^n, using binomial coefficients.
Substitution – The process of replacing variables with their given values in an expression.
Combinatorics – Understanding and applying combinations, represented as C(n, k), to determine the coefficients in the expansion.