Question

Find the indicated term of each expansion.

Third term of ${\left(x+y\right)}^8$

Short answer

The third term of the expansion is$T_3=28x^6{y}^2$

Step-by-step solution

Step 1. Given Information.

Given, binomial expansion is ${\left(x+y\right)}^8$ and the third term of the expansion is to be determined.

Step 2. Calculation.

The general term of the binomial expansion ${\left(a+b\right)}^n$ is given by:

$T_{r+1}=\frac{n!}{\left(n-r\right)!r!}{a}^{n-r}{b}^r$

Here n=8, a = x, b = y and r + 1 = 3.

Plugging the values:

$\begin{array}{l}{T}_{r+1}=\frac{n!}{\left(n-r\right)!r!}{a}^{n-r}{b}^r\\ T_3=\frac{8!}{\left(8-2\right)!2!}{x}^{8-2}{y}^2\\ T_3=\frac{8!}{6!2!}{x}^6{y}^2\\ T_3=28x^6{y}^2\end{array}$

Step 3. Conclusion.

Hence, the third term of the expansion is $T_3=28x^6{y}^2$.