Question

Find the term independent of x in the expansion of the expression: 

(vii)  \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\]

 

Answer 1

(vii)  Suppose the (r + 1)th term in the given expression is independent of x.
Now, 

\[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8 \]
\[ T_{r + 1} = ^{8}{}{C}_r \left( \frac{1}{2} x^{1/3} \right)^{8 - r} ( x^{- 1/5} )^r \]
\[ =^{8}{}{C}_r . \frac{1}{2^{8 - r}} x^\frac{8 - r}{3} - \frac{r}{5} \]
\[\text{ For this term to be independent of x, we must have } \]
\[\frac{8 - r}{3} - \frac{r}{5} = 0\]
\[ \Rightarrow 40 - 5r - 3r = 0\]
\[ \Rightarrow 8r = 40\]
\[ \Rightarrow r = 5\]
\[\text{ Hence, the required term is the 6th term } . \]
\[\text{ Now, we have: } \]
\[ ^{8}{}{C}_5 \times \frac{1}{2^{8 - 5}}\]
\[ = \frac{8 \times 7 \times 6}{3 \times 2 \times 8} = 7\]