Finding The 8th Term Of (√x + 2/√x)^12 A Step-by-Step Guide
Introduction
In this article, we delve into the fascinating world of binomial expansions. Our main goal is to find the 8th term in the expansion of the expression (√x + 2/√x)^12. This problem elegantly combines algebraic manipulation with the binomial theorem, providing a great exercise in understanding polynomial expansions. To solve this, we'll leverage the binomial theorem, a powerful tool that allows us to expand expressions of the form (a + b)^n. This theorem is not only crucial in algebra but also has applications in various fields such as statistics, probability, and even computer science. Before diving into the specifics of the 8th term, let's first understand the binomial theorem and its components, which will lay the foundation for our solution. We'll then break down the given expression and carefully apply the theorem to pinpoint the exact term we're looking for. Our journey will involve understanding binomial coefficients, identifying the general term in a binomial expansion, and substituting the appropriate values to arrive at our final answer. By the end of this article, you'll not only know how to find a specific term in a binomial expansion but also appreciate the underlying principles that make it possible. So, let’s embark on this mathematical adventure and uncover the intricacies of binomial expansion together!
Understanding the Binomial Theorem
The binomial theorem provides a systematic way to expand expressions of the form (a + b)^n, where n is a non-negative integer. This theorem is a cornerstone of algebra and is expressed mathematically as:
(a + b)^n = Σ [nCk * a^(n-k) * b^k], where k ranges from 0 to n
Here, the symbol nCk represents the binomial coefficient, also known as "n choose k", which is the number of ways to choose k elements from a set of n elements. It is calculated using the formula:
nCk = n! / (k! * (n-k)!)
where "!" denotes the factorial function (e.g., 5! = 5 × 4 × 3 × 2 × 1). The binomial coefficient plays a crucial role in determining the numerical coefficient of each term in the expansion. The exponents (n-k) and k dictate the powers of the terms 'a' and 'b' in each term of the expansion, respectively. Understanding how these components interact is crucial for successfully applying the binomial theorem. To illustrate, let's consider a simple example: expanding (x + y)^3. Using the binomial theorem, we would have four terms, corresponding to k = 0, 1, 2, and 3. Each term involves a binomial coefficient, a power of x, and a power of y. By calculating these components and summing them up, we obtain the expanded form of the expression. This fundamental understanding of the binomial theorem and its components—binomial coefficients, exponents, and the summation process—is essential for tackling more complex problems, such as finding a specific term in a binomial expansion.
Identifying the General Term
In the context of finding a specific term within a binomial expansion, the concept of the general term is invaluable. The general term, often denoted as Tk+1, provides a formula for calculating any term in the expansion without having to expand the entire expression. This is particularly useful when dealing with higher powers or when the specific term we seek is located further into the expansion. The general term formula is derived directly from the binomial theorem and is given by:
Tk+1 = nCk * a^(n-k) * b^k
where:
- Tk+1 represents the (k+1)-th term in the expansion.
- n is the exponent in the binomial expression (a + b)^n.
- k is the term number, starting from 0.
- nCk is the binomial coefficient, as previously defined.
- a and b are the terms within the binomial expression.
To effectively use the general term formula, it's crucial to correctly identify the values of n, k, a, and b from the given expression. For instance, if we are looking for the 5th term in the expansion of (2x - 3y)^8, then n would be 8, k would be 4 (since we are looking for the 5th term, k = 5 - 1), a would be 2x, and b would be -3y. Substituting these values into the formula allows us to calculate the 5th term directly, without needing to expand the entire expression. This not only saves time but also reduces the chances of making errors during the expansion process. The ability to identify and apply the general term formula is a powerful technique in binomial expansions, enabling us to efficiently target specific terms and solve a wide range of problems.
Applying the Binomial Theorem to (√x + 2/√x)^12
Now, let's apply our knowledge to the specific problem at hand: finding the 8th term in the expansion of (√x + 2/√x)^12. We will use the general term formula we discussed earlier:
Tk+1 = nCk * a^(n-k) * b^k
First, we need to identify the values of n, a, and b from the given expression. In this case:
- n = 12 (the exponent)
- a = √x (the first term)
- b = 2/√x (the second term)
Since we are looking for the 8th term, k will be 7 (because Tk+1 represents the (k+1)-th term, so for the 8th term, k = 8 - 1 = 7). Now we have all the pieces needed to plug into the general term formula. Substituting the values, we get:
T8 = 12C7 * (√x)^(12-7) * (2/√x)^7
This expression now needs to be simplified. The next step is to calculate the binomial coefficient 12C7 and simplify the powers of √x and 2/√x. This will involve understanding how to handle fractional exponents and applying the properties of exponents. The simplification process is crucial to arriving at the correct final term. By carefully breaking down each component and applying the relevant mathematical rules, we can effectively determine the 8th term of the expansion. This step-by-step approach not only helps us solve the problem at hand but also reinforces our understanding of the underlying principles of binomial expansions and algebraic manipulations.
Calculating the 8th Term
Let's proceed with the calculation of the 8th term. We have the expression:
T8 = 12C7 * (√x)^5 * (2/√x)^7
First, let's calculate the binomial coefficient 12C7:
12C7 = 12! / (7! * 5!) = (12 × 11 × 10 × 9 × 8) / (5 × 4 × 3 × 2 × 1) = 792
Now, let's simplify the powers of √x and 2/√x. Recall that √x can be written as x^(1/2). So,
(√x)^5 = (x(1/2))5 = x^(5/2)
and
(2/√x)^7 = 2^7 / (√x)^7 = 128 / (x(1/2))7 = 128 / x^(7/2)
Substituting these values back into the expression for T8, we get:
T8 = 792 * x^(5/2) * (128 / x^(7/2))
Now, we can simplify further by multiplying the numerical coefficients and combining the powers of x:
T8 = 792 * 128 * (x^(5/2) / x^(7/2))
T8 = 101376 * x^(5/2 - 7/2)
T8 = 101376 * x^(-2/2)
T8 = 101376 * x^(-1)
T8 = 101376 / x
Therefore, the 8th term in the expansion of (√x + 2/√x)^12 is 101376 / x. This step-by-step calculation demonstrates how each component of the general term formula contributes to the final result. From calculating the binomial coefficient to simplifying the exponents, each step requires careful attention to detail. The final simplification of the expression highlights the power of algebraic manipulation in arriving at a concise and accurate answer.
Conclusion
In conclusion, we have successfully found the 8th term in the expansion of (√x + 2/√x)^12 to be 101376 / x. This journey through binomial expansion has highlighted the importance of understanding and applying the binomial theorem. We started by introducing the binomial theorem and its components, emphasizing the significance of binomial coefficients and the general term formula. We then applied the general term formula to our specific problem, carefully identifying the values of n, a, b, and k. The calculation process involved simplifying binomial coefficients and manipulating exponents, ultimately leading us to the correct answer. This problem serves as a great example of how algebraic concepts can be combined to solve complex problems. The binomial theorem is a powerful tool in mathematics, with applications extending beyond simple expansions. Mastering this theorem not only enhances one's algebraic skills but also provides a foundation for more advanced topics in mathematics and related fields. We hope this detailed explanation has provided a clear understanding of how to find specific terms in binomial expansions and has inspired you to explore further the fascinating world of mathematics.
