Problem 31 - Algebra and Trigonometry

Question

Find 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 \(25 a^{26/3}\) and \(a^{25/3}\).

Step 1: Understand the Binomial Expansion

We are given the expression \((a^{2/3} + a^{1/3})^{25}\). To find any specific term in the expansion, we use the binomial theorem: \[(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]. In our case, \(x = a^{2/3}\), \(y = a^{1/3}\), and \(n = 25\).

Step 2: Determine the General Term

The general term \(T_k\) in the expansion is given by \(T_k = \binom{n}{k} (a^{2/3})^{n-k} (a^{1/3})^k\). For our expression: \[T_k = \binom{25}{k} (a^{2/3})^{25-k} (a^{1/3})^k = \binom{25}{k} a^{(2/3)(25-k) + (1/3)k}\]. Simplify the exponent of \(a\): \[ = \binom{25}{k} a^{50/3 - (2/3)k + (1/3)k} = \binom{25}{k} a^{(50-k)/3}\].

Step 3: Find the Last Two Terms

To find the last two terms, we need the terms with the smallest powers of \(a\), which would be when \(k\) is the largest because the exponent \((50-k)/3\) decreases as \(k\) increases. Let's compute for \(k=24\) and \(k=25\).

Step 4: Compute for k = 24

For \(k = 24\): \[T_{24} = \binom{25}{24} a^{(50-24)/3} = 25 a^{26/3}\] because \(\binom{25}{24} = 25\).

Step 5: Compute for k = 25

For \(k = 25\): \[T_{25} = \binom{25}{25} a^{(50-25)/3} = 1 \cdot a^{25/3} = a^{25/3}\] because \(\binom{25}{25} = 1\).

Step 6: Conclusion for Last Two Terms

The last two terms of the expansion are \[25 a^{26/3}\] and \[a^{25/3}\].

Key Concepts

Binomial Expansion

The Binomial Expansion is a fundamental concept in algebra used to expand expressions of the form \((x + y)^n\). This is expressed using the Binomial Theorem, which provides us with a formula to expand and calculate each specific term in the sequence. In essence, it breaks down the process of raising a binomial to a power into a series of terms. Each term has the general form \( \binom{n}{k} x^{n-k} y^k \), where \(\binom{n}{k}\) are the binomial coefficients and \(n\) is the power to which the binomial is raised.
For example, in the binomial expansion of \((a^{2/3} + a^{1/3})^{25}\), each term follows this principle. With each increase in \(k\), we adjust the powers of \(x\) and \(y\). The sum of the powers in each term always equals \(n\). This methodical approach allows us to manage complex polynomials effectively and find specific terms without fully expanding the entire expression.

Binomial Coefficients

Binomial Coefficients are the numerical factors that multiply the terms in a binomial expansion. They are denoted as \(\binom{n}{k}\), which is read as 'n choose k'. This notation is derived from combinatorics, where it represents the number of ways to choose \(k\) items from \(n\) items without regard to the order. The formula to calculate a binomial coefficient is given by:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Here, \(!\) denotes the factorial, which is the product of all positive integers up to that number.
In our specific example of expanding \((a^{2/3} + a^{1/3})^{25}\), we see binomial coefficients used multiple times, like \(\binom{25}{24} = 25\) and \(\binom{25}{25} = 1\). These coefficients are key in determining the weight or "size" of each term in the expansion.

Exponentiation

Exponentiation is the mathematical operation involving exponents. It is a shortcut for multiplying a number by itself a certain number of times. In the context of binomial expansion, exponentiation defines the power structure of each term in the sequence.
For instance, in the expansion of \((a^{2/3} + a^{1/3})^{25}\), each term is a result of exponentiation applied both to individual components \(a^{2/3}\) and \(a^{1/3}\), combined with the power raised, which is 25 overall.
The general form inside a term is \(x^{n-k}y^k\). Simplifying these powers properly is crucial. For higher algebraic manipulations, it helps in understanding the distribution of terms and calculating specific values throughout the expansion.

• Exponents show how many times a base is used in a multiplication.
• Fractional exponents, like \(a^{2/3}\), mean taking roots, where the denominator indicates the root and the numerator shows the power to be applied afterward.

Understanding exponentiation within binomial expansions leads to a better grasp of how terms interact and the overall behavior of polynomial functions.