Find the term independent of $x$ in the expansion of $\left(\sqrt[3]{x}+\frac{1}{2 \sqrt[3]{x}}\right)^{18}, x\,>\,0$
Find the term independent of $x$ in the expansion of $\left(\sqrt[3]{x}+\frac{1}{2 \sqrt[3]{x}}\right)^{18}, x\,>\,0$
We have ${T_{r + 1}} = {\,^{18}}{C_r}{(\sqrt[3]{x})^{18 - r}}{\left( {\frac{1}{{2\sqrt[3]{x}}}} \right)^r}$
$ = {\,^{18}}{C_r}{x^{\frac{{18 - r}}{3}}} \cdot \frac{1}{{{2^r} \cdot {x^{\frac{r}{3}}}}} = {\,^{18}}{C_r}\frac{1}{{{2^r}}} \cdot {x^{\frac{{18 - 2r}}{3}}}$
Since we have to find a term independent of $x$, i.e., term not having $x$, so take $\frac{18-2 r}{3}=0$
We get $r=9 .$ The required term is ${\,^{18}}{C_9}\frac{1}{{{2^9}}}$
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