Understanding the Fourth Term of Binomial Expansion of 2y^1/2
Understanding the Fourth Term of Binomial Expansion of 2y^1/2
When dealing with binomial expansions, especially with fractional exponents, it is essential to apply the Binomial Theorem correctly to identify the specific terms. In this article, we will explore the fourth term of the binomial expansion of 2y1/2. We will break down the steps and explain the mathematical process involved.
Introduction to Binomial Expansion
The Binomial Theorem is a powerful tool for expanding powers of binomials in the form (a b)n. The general formula for expansion is:
(a b)n Σ[k0 to n] [nCk * an-k * bk]
Applying Binomial Expansion to 2y1/2
Given the expression 2y1/2, we need to express it as a binomial fraction. In this case, the expression can be written as (21y/2)1/2.
Step-by-Step Breakdown
1. **Express the given expression as a binomial:** 2y1/2 (21y/2)1/2 2. **Break down the binomial into a more familiar form:** (21y/2)1/2 (21/2 * (1y/2))1/2 3. **Apply the binomial expansion formula:** (a b)n Σ[k0 to n] [nCk * an-k * bk] 4. **Identify the variables and constants: a 21/2, b 1y/2, n 1/2** 5. **Calculate the fourth term (k3):** The fourth term in a binomial expansion corresponds to the term where k3. Using the general term formula, we get:
T4 [1/2C3] * (21/2)1/2-3 * (1y/2)3Calculating the Fourth Term
To calculate the fourth term accurately, we need to use the combination formula (nCk) and simplify the expression:
T4 [1/2C3] * (21/2)-5/2 * (1/2)3 * y3
1. **Calculate the combination (1/2C3):**
(1/2C3) (1/2) * (1/2-1) * (1/2-2) / (3!) (1/2) * (-1/2) * (-3/2) / 6 -1/16
2. **Simplify the expression: (21/2)-5/2:**
(21/2)-5/2 2-5/4 1/25/4 1/√(32)
3. **Combine all the terms: (1/16) * (1/32) * y3:**
T4 (1/16) * (1/32) * y3 1/512 * y3 √2/128 * y3
Conclusion
Thus, the fourth term of the binomial expansion of 2y1/2 is √2/128 * y3. This process demonstrates the application of the Binomial Theorem to fractional exponents and the importance of careful calculation. Understanding these steps is crucial for solving complex expansions and for improving skills in algebra.