Question

In the expansion of ${(\frac{1}{2}{x}^{1/3}+x^{-1/5})}^8$, the term independent of x is

• A. $T_5$
• B. $T_6$
• C. $T_7$
• D. $T_8$

Answer 1

The correct option is B

$T_6$

$T_6$

Suppose the (r+1)th term in the given expansion is independent of x.

Then, we have:

$T_{r+1}={}^8{C}_r{\left(\frac{1}{2}{x}^{\frac{1}{3}}\right)}^{8-r}{\left(x^{-\frac{1}{5}}\right)}^r$

$={}^8{C}_r\frac{1}{2^{8-r}}{x}^3-\frac{r}{5}$

For this term to be independent of x, we must have:

$\frac{8-r}{3}-\frac{r}{5}=0$

$\Rightarrow 40-5r-3r=0$

$\Rightarrow r=5$

Hence, the required term is the 6th term, i.e.$T_6.$