In Exercises 39–48, find the term indicated in each expansion. (x²+y³)^8; sixth term

Verified step by step guidance

1. Identify the general term in the binomial expansion of \((x^2 + y^3)^8\) using the Binomial Theorem, which is given by \(T_{k+1} = \binom{n}{k} (a)^{n-k} (b)^k\).
2. In this problem, \(n = 8\), \(a = x^2\), and \(b = y^3\).
3. To find the sixth term, set \(k = 5\) because the term number is \(k+1\).
4. Substitute \(n = 8\), \(k = 5\), \(a = x^2\), and \(b = y^3\) into the formula: \(T_6 = \binom{8}{5} (x^2)^{8-5} (y^3)^5\).
5. Simplify the expression: \(T_6 = \binom{8}{5} (x^2)^3 (y^3)^5\).

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)^n. It states that the expansion can be expressed as a sum of terms involving binomial coefficients, which can be calculated using combinations. Each term in the expansion is of the form C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or 'n choose k', represent the number of ways to choose k elements from a set of n elements without regard to the order of selection. They are calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial. These coefficients play a crucial role in determining the coefficients of each term in the binomial expansion.

Term in a Binomial Expansion

In a binomial expansion, each term corresponds to a specific value of k, where k ranges from 0 to n. The general term can be expressed as T(k+1) = C(n, k) * a^(n-k) * b^k. To find a specific term, such as the sixth term, one must identify the appropriate value of k and substitute it into the general term formula, ensuring to account for the indexing starting from zero.