Calculate the term independent of x in the expansion of (x/2 - 3)^6.

Calculate the term independent of x in the expansion of (x/2 - 3)^6.

Step 1: Identify the expression to expand, which is (x/2 - 3)^6.
Step 2: Use the Binomial Theorem to expand the expression. The general term in the expansion is given by C(n, k) * (a)^(n-k) * (b)^k, where n is the exponent, a is the first term, b is the second term, and k is the term number.
Step 3: In our case, n = 6, a = (x/2), and b = (-3). So the general term is C(6, k) * (x/2)^(6-k) * (-3)^k.
Step 4: We want to find the term that does not contain x, which means we need to set the exponent of x to 0. This happens when (6-k) = 0, or k = 6.
Step 5: However, we need to find the term independent of x, which occurs when k = 3 (since 6 - 3 = 3).
Step 6: Calculate the coefficient for k = 3: C(6, 3) = 20.
Step 7: Calculate (x/2)^3 = (1/8)x^3, and (-3)^3 = -27.
Step 8: Combine these values: 20 * (1/8) * (-27).
Step 9: Simplify the expression: 20 * (1/8) = 2.5, and then 2.5 * (-27) = -67.5.
Step 10: The term independent of x in the expansion is -67.5.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the binomial theorem.
Finding the Constant Term – Identifying the term in the expansion that does not contain the variable x.
Combinatorial Coefficients – Using binomial coefficients C(n, k) to determine the number of ways to choose k elements from n.