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 \(\left(x^{2} + y^{3}\right)^8\). The general term is given by the formula: \(T_{k+1} = \binom{8}{k} (x^{2})^{8-k} (y^{3})^{k}\), where \(k\) ranges from 0 to 8.
2. Since we want the sixth term, set \(k + 1 = 6\), which means \(k = 5\).
3. Substitute \(k = 5\) into the general term formula: \(T_6 = \binom{8}{5} (x^{2})^{8-5} (y^{3})^{5}\).
4. Simplify the exponents: \((x^{2})^{3} = x^{6}\) and \((y^{3})^{5} = y^{15}\), so the term becomes \(\binom{8}{5} x^{6} y^{15}\).
5. Calculate the binomial coefficient \(\binom{8}{5}\) using the formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), then write the full expression for the sixth term.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n into a sum of terms involving binomial coefficients. Each term is given by C(n, k) * a^(n-k) * b^k, where C(n, k) is a combination representing the number of ways to choose k elements from n.

Binomial Coefficients and Combinations

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to select k items from n without regard to order. They are calculated using factorials: C(n, k) = n! / (k! * (n-k)!). These coefficients determine the coefficients of terms in a binomial expansion.

Term Identification in Binomial Expansion

In the expansion of (a + b)^n, the (k+1)-th term corresponds to C(n, k) * a^(n-k) * b^k. To find a specific term, identify k based on the term number, then substitute into the formula. For example, the sixth term corresponds to k = 5.