Problem 49 - Precalculus with Limits

Question

In Exercises 45 - 52, find the specified \( n \)th term in the expansion of the binomial. \( \left(4x + 3y\right)^{9}, \quad n = 8 \)

Step-by-Step Solution

Answer

The 8th term in the binomial expansion of \((4x + 3y)^9\) is \(13749310575y^8x\)

Step 1 - Identify a, b, n, and k

First, the terms of the binomial, \(a\) and \(b\), need to be identified. Here, \(a = 4x\), \(b = 3y\), \(n = 9\) and since the term to be found is the 8th, \(k = n - 1 = 8\)

Step 2 - Apply the Formula

Now, plug \(a = 4x\), \(b = 3y\), \(n = 9\), and \(k = 8\) into the formula \(T_n = ^nC_k \cdot (a)^{n-k} \cdot (b)^k\)

Step 3 - Simplify the Term

Solve the combination, and simplify the terms to find out the 8th term in the expansion.

Step 4 - Final Result

The final step is to simplify the combination to get the 8th term in the binomial expansion.

Key Concepts

Understanding the nth Term

The "nth term" in the context of a binomial expansion refers to any specific term within the expanded form of a binomial expression raised to a power. In binomial expansions, each term is systematically generated by plugging into the binomial theorem formula. For example, if you are asked to find the 8th term, it means you're specifically looking at one particular term among the expanded set.
To retrieve this specific term, you use the formula for the nth term of a binomial expansion:

• Identify the base terms of the binomial, which are typically represented as \(a\) and \(b\).
• Determine the power \(n\), which is the exponent the binomial is raised to.
• Identify \(k\), where \(k = n - 1\) for the nth term.
• Apply the formula: \[ T(n + 1) = \binom{n}{k} \cdot a^{n-k} \cdot b^k \]

Using these steps ensures that you can accurately pinpoint any desired term within the expansion.

Exploring Binomial Expansion

The concept of "binomial expansion" involves expanding an expression that is raised to a large power, such as \((a + b)^n\). Binomial expansion converts this compact expression into a sum of terms, making it clearer. This is particularly useful for higher powers, where expanding manually would be tedious.
The binomial theorem provides a formula that tells us:

• Each term in the expansion includes a coefficient, a product of powers of \(a\), and \(b\), specified by the combination formula \(\binom{n}{k}\).
• This coefficient is determined by combinations from combinatorics, which dictates how many ways \(k\) elements can be selected from \(n\) elements.
• The expanded form will have \((n + 1)\) terms, starting from \((a^n)\) ending with \((b^n)\).

Leveraging the binomial theorem simplifies calculations and analysis and helps avoid computational errors.

The Role of Combinatorics

Combinatorics, a field of mathematics focusing on counting, plays a crucial role in binomial expansion, especially when determining coefficients of each term. Every coefficient in a binomial expansion is calculated using combinations, represented by \(\binom{n}{k}\), symbolizing the "choose" operation.
Here's how it works:

• \(\binom{n}{k}\) is read as "n choose k," indicating the number of ways to choose \(k\) items from \(n\) items without regard to order.
• The formula for combinations is given by \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
• In the context of binomial expansions, \(\binom{n}{k}\) determines how often specific terms in the expansion will appear.

This mathematical principle ensures every term in the expansion has the correct coefficient, enabling the accurate reconstruction of the expanded expression.