Calculate the term independent of x in the expansion of (x/2 - 3)^8.

Calculate the term independent of x in the expansion of (x/2 - 3)^8.

Step 1: Identify the expression to expand, which is (x/2 - 3)^8.
Step 2: Use the Binomial Theorem to expand the expression. The general term in the expansion is given by C(n, k) * (a)^k * (b)^(n-k), where n is the exponent, k is the term number, a is the first term, and b is the second term.
Step 3: In our case, n = 8, a = (x/2), and b = (-3). So the general term is C(8, k) * (x/2)^k * (-3)^(8-k).
Step 4: We want to find the term that is independent of x. This occurs when the power of x is zero. The power of x in the term is k, so we set k = 0.
Step 5: To find the term independent of x, we need to find k such that (x/2)^k has no x. This happens when k = 4 because (x/2)^4 = (x^4)/(2^4) and we want the x part to cancel out.
Step 6: Calculate the coefficient for k = 4. The term is C(8, 4) * (x/2)^4 * (-3)^(8-4).
Step 7: Calculate C(8, 4), which is 70.
Step 8: Calculate (x/2)^4, which is (1/16) because (x^4)/(2^4) = (x^4)/(16).
Step 9: Calculate (-3)^4, which is 81.
Step 10: Combine these values: 70 * (1/16) * 81 = 70 * 81 / 16 = 5670 / 16 = 256.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the binomial theorem.
Finding the Term Independent of a Variable – Identifying the specific term in a binomial expansion that does not contain the variable x.
Combinatorial Coefficients – Using combinations to determine the coefficients in the binomial expansion.