Problem 58 - Beginning and Intermediate Algebra

Question

Find the indicated term of each binomial expansion. \((2 w-1)^{9} ;\) seventh term

Step-by-Step Solution

Answer

The seventh term in the binomial expansion of \((2w - 1)^9\) is \(672w^3\).

Step 1: Identify \(a\), \(b\), \(n\), and \(k\)

In this problem, we have the binomial expansion \((2w - 1)^9\). We can identify \(a = 2w\), \(b = -1\), and \(n = 9\). We want to find the seventh term, so according to the indexing of the sum from the binomial theorem, we need to find the term corresponding to \(k = 6\).

Step 2: Apply the Binomial Theorem using the chosen \(k\) value

We will now apply the Binomial Theorem formula for \(k=6\). Term with \(k = 6\): \(\binom{n}{k} a^{n-k} b^{k}\) Substitute the values \(a = 2w\), \(b = -1\), \(n = 9\), and \(k = 6\): Term with \(k = 6\): \(\binom{9}{6} (2w)^{9-6} (-1)^{6}\)

Step 3: Compute the binomial coefficient, exponentiation, and simplify the term

Calculate the binomial coefficient, exponentiate the base, and simplify the term. Binomial coefficient: \(\binom{9}{6} = \dfrac{9!}{6!(9-6)!} = \dfrac{9!}{6!3!} = \dfrac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84\) Exponentiation: \((2w)^{9-6} = (2w)^3 = 8w^3\) The term simplifies to: \(84 \times 8w^3 \times (-1)^6 = 84 \times 8w^3 = 672w^3\) Therefore, the seventh term in the binomial expansion of \((2w - 1)^9\) is \(672w^3\).