Find the coefficient of x^0 in the expansion of (2x + 3)^4.

Find the coefficient of x^0 in the expansion of (2x + 3)^4.

Step 1: Understand that x^0 means we want the constant term in the expansion, which does not contain x.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = Σ (C(n, k) * a^(n-k) * b^k) for k = 0 to n.
Step 3: In our case, a = 2x, b = 3, and n = 4.
Step 4: To find the coefficient of x^0, we need to set (2x)^k to 0, which means k must be 0 (since any power of x greater than 0 will not give us x^0).
Step 5: Calculate the term when k = 0: C(4, 0) * (2x)^0 * 3^4.
Step 6: C(4, 0) is 1 (the number of ways to choose 0 items from 4).
Step 7: (2x)^0 is also 1 (anything to the power of 0 is 1).
Step 8: Now calculate 3^4, which is 3 * 3 * 3 * 3 = 81.
Step 9: Multiply the results: 1 * 1 * 81 = 81.
Step 10: Therefore, the coefficient of x^0 in the expansion of (2x + 3)^4 is 81.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Coefficient Extraction – Identifying the coefficient of a specific term (in this case, x^0) in the expanded form.