Problem 31 - Precalculus: Mathematics for Calculus

Question

Find the indicated terms in the expansion of the given binomial. The last two terms in the expansion of \(\left(a^{2 / 3}+a^{1 / 3}\right)^{25}\).

Step-by-Step Solution

Answer

The last two terms are \(25 a^{\frac{26}{3}}\) and \(a^{\frac{25}{3}}\).

Step 1: Recall Binomial Theorem

The Binomial Theorem states that any binomial raised to a power can be expanded into a series: \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \)where \(\binom{n}{k}\) is the binomial coefficient.

Step 2: Define Variables

For the binomial \((a^{2/3} + a^{1/3})^{25}\), let \(x = a^{2/3}\) and \(y = a^{1/3}\) and \(n = 25\). Then, each term in the expansion is given by \(\binom{25}{k} (a^{2/3})^{25-k} (a^{1/3})^k\).

Step 3: Simplify General Term

The general term in the expansion is \[\binom{25}{k} (a^{2/3})^{25-k} (a^{1/3})^k = \binom{25}{k} a^{\frac{2}{3}(25-k) + \frac{1}{3}k}\]Simplifying the exponent of \(a\) gives \[a^{\frac{2}{3}(25-k) + \frac{1}{3}k} = a^{\frac{50 - 2k + k}{3}} = a^{\frac{50 - k}{3}}\].

Step 4: Determine Exponent for the Last Two Terms

The exponent of \(a\) decreases as \(k\) increases, so the last two terms occur when \(k = 24\) and \(k = 25\). Calculate the exponents for these terms: - When \(k = 24\), the exponent is \(\frac{50-24}{3} = rac{26}{3}\)- When \(k = 25\), the exponent is \(\frac{50-25}{3} = rac{25}{3}\).

Step 5: Calculate Last Two Terms

Substitute \(k = 24\) and \(k = 25\) into the general term:- For \(k = 24\), the term is \(\binom{25}{24} a^{\frac{26}{3}} = 25 a^{\frac{26}{3}}\).- For \(k = 25\), the term is \(\binom{25}{25} a^{\frac{25}{3}} = 1 \cdot a^{\frac{25}{3}} = a^{\frac{25}{3}}\).

Key Concepts

Binomial Coefficient

The binomial coefficient is a key component of the Binomial Theorem that helps determine the number of ways to choose items from a larger set. In the context of the theorem, it represents the number of terms in an expansion of a binomial expression, using the notation \( \binom{n}{k} \). This notation is read as "n choose k," indicating the number of combinations possible when selecting \(k\) items from \(n\) items without regard to order.

To compute the binomial coefficient, the formula is:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

Where \(n!\) (read as "n factorial") is the product of all positive integers up to \(n\). For instance, the binomial coefficient \( \binom{25}{24} \) is used to calculate one of the last terms in the expansion of \((a^{2/3} + a^{1/3})^{25}\), which simplifies to 25. This is because selecting 24 items from 25 in all possible ways is essentially the same as choosing the last remaining item.Understanding the binomial coefficient is crucial for determining the specific coefficients of terms within a binomial expansion.

Exponent Simplification

Exponent simplification is a mathematical process used to make exponents easier to work with, especially in algebraic expressions. When expanding a binomial like \((a^{2/3} + a^{1/3})^{25}\), simplifying the exponents in each term is vital to determine the correct power of the terms.

For the general term \( \binom{25}{k} (a^{2/3})^{25-k} (a^{1/3})^k \), it’s necessary to simplify the expression inside the exponent. This involves combining the separate exponents of \(a\):

• \( (a^{2/3})^{25-k} = a^{\frac{2}{3}(25-k)} \)
• \( (a^{1/3})^k = a^{\frac{1}{3}k} \)

When you combine these:

• \( a^{\frac{2}{3}(25-k) + \frac{1}{3}k} \)
• \( a^{\frac{50-2k+k}{3}} = a^{\frac{50-k}{3}} \)

By simplifying the exponent, you determine the exact power of \(a\) in each term of the expansion. This was essential for identifying the specific terms when \(k = 24\) and \(k = 25\), giving exponents like \(\frac{26}{3}\) and \(\frac{25}{3}\) respectively.

Binomial Expansion Steps

The Binomial Expansion process involves expanding expressions raised to a power using the Binomial Theorem. This theorem provides a structured method to express the powers of binomials as sums of terms involving binomial coefficients.

Here’s a simple guide through the steps of binomial expansion:

• Identify the binomial expression and the power it is raised to. For example, in our task, it's \((a^{2/3} + a^{1/3})^{25}\).
• Apply the Binomial Theorem: \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \).
• Define the variables: Here, let \(x = a^{2/3}\), \(y = a^{1/3}\), and \(n = 25\).
• Calculate each term in the expansion using the binomial coefficients and simplified exponents, \( \binom{n}{k} x^{n-k} y^k \).
• Let's wrap it up by finding the specific terms, especially when focusing on the last ones (\(k = 24\) and \(k = 25\)). For this, the exponents of \(a\) decrease with an increasing \(k\), allowing you to pinpoint the last two terms.

Following these steps allows you to break down the expansion into manageable parts, simplifying the process of finding any specific term within the binomial.