Question

The number of irrational terms in the expansion of ${\left(4^{1/5}+7^{1/10}\right)}^{45}$ is
(a) 40
(b) 5
(c) 41
(d) none of these

Answer 1

(c) 41

$$\begin{gathered} \mathrm{The}\mathrm{general}\mathrm{term}{T}_{r+1}\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{expansion}\mathrm{is}\mathrm{given}\mathrm{by} \\ {}^{45}C_r(4^{1/5}{)}^{45-r}(7^{1/10}{)}^r \\ \mathrm{For}{T}_{r+1}\mathrm{to}\mathrm{be}\mathrm{an}\mathrm{integer},\mathrm{we}\mathrm{must}\mathrm{have}\frac{r}{5}\mathrm{and}\frac{r}{10}\mathrm{as}\mathrm{integers}\mathrm{i}.\mathrm{e}.0\le r\le 45 \\ \therefore r=0,10,20,30\mathrm{and}40 \\ \mathrm{Hence},\mathrm{there}\mathrm{are}5\mathrm{rational}\mathrm{and}41,\mathrm{i}.\mathrm{e}.,46-5,\mathrm{irrational}\mathrm{terms}. \end{gathered}$$