Find the indicated term of the binomial expansion.
-$\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 13 }$; eighth term
A) $- 329,472 x ^ { 15 } y ^ { 16 }$
B) $- 3,432 x ^ { 18 } y ^ { 14 }$
C) $- 2,574 x ^ { 15 } y ^ { 16 }$
D) $- 219,648 x ^ { 18 } y ^ { 14 }$

Question

Quizplus · Accepted Answer

The eighth term of the binomial expansion $(a - b)^n$ can be found using the formula for the $k$th term: $T_k = \binom{n}{k-1} a^{n-k+1} b^{k-1}$. For the eighth term ($k = 8$) of $(x^3 - 2y^2)^{13}$, we have $n = 13$, $a = x^3$, and $b = -2y^2$. Thus, $T_8 = \binom{13}{7} (x^3)^{13-7} (-2y^2)^7 = \binom{13}{7} x^{18} (-2)^7 y^{14}$. Calculating $\binom{13}{7} = 1716$ and $(-2)^7 = -128$, we get $T_8 = 1716 \times -128 x^{18} y^{14} = -219,648 x^{18} y^{14}$.

Find the Indicated Term of the Binomial Expansion (x3−2y2)13\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 13 }(x3−2y2)13

Find the Indicated Term of the Binomial Expansion $\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 13 }$