Question

Find the term independent of $x$ in the expansion of ${(\frac{\sqrt{x}}{3}+\frac{3}{2x^2})}^{10}$.

Answer 1

$Generalterm{T}_{r+1}{=}^{10}{C}_r{\left(\right(\frac{\sqrt{}x}{3}\left)\right)}^r{\left(\right(\frac{3}{2x^2}\left)\right)}^{10-r}{=}^{10}{C}_r\left(\frac{x^{\left(\frac{r}{2}\right)}}{3^{\left(\frac{r}{2}\right)}}\right)\left(\frac{3^{10-r}}{2^{10-r}{x}^{20-2r}}\right){=}^{10}{C}_r{3}^{(10-r-r/2)}{2}^{r-10}{x}^{(r/2-20+2r)}nowwewanttermindependentofx\therefore x^{r/2-20+2r}=x^{0}or(\frac{5r}{2}-20)=0or\left(\frac{5r}{2}\right)=20\therefore r=\left(\frac{40}{5}\right)=8\therefore term=T_{8+1}=T_9$