Question

In the expansion of ${\left(\frac{1}{2}{x}^{1/3}+x^{-1/5}\right)}^8$, the term independent of x is
(a) T₅
(b) T₆
(c) T₇
(d) T₈

Answer 1

(b) T₆
Suppose the (r + 1)th term in the given expansion is independent of x.
Thus, we have:
$$\begin{gathered} T_{r+1}={}^{8}C_r{\left(\frac{1}{2}{x}^{1/3}\right)}^{8-r}(x^{-1/5}{)}^r \\ ={}^{8}C_r\frac{1}{2^{8-r}}{x}^{\frac{8-r}{3}-\frac{r}{5}} \\ \mathrm{For}\mathrm{this}\mathrm{term}\mathrm{to}\mathrm{be}\mathrm{independent}\mathrm{of}\mathit{}x,\mathrm{we}\mathrm{must}\mathrm{have} \\ \frac{8-r}{3}-\frac{r}{5}=0 \\ \Rightarrow 40-5r-3r=0 \\ \Rightarrow r=5 \\ \mathrm{Hence},\mathrm{the}\mathrm{required}\mathrm{term}\mathrm{is}\mathrm{the}6\mathrm{th}\mathrm{term},\mathrm{i}.\mathrm{e}.T_6 \end{gathered}$$