Question

The number of rational terms in the expansion of ${(\sqrt[4]{45}+\sqrt[5]{54})}^{100}$ is

• A. $50$
• B. $5$
• C. $6$
• D. $51$

Answer 1

Answer: (C)

$6$

$(4\sqrt{5}+5\sqrt{4}{)}^{100}$

$T_{r+1}={100}_{cr}{\left(5\frac{1}{4}\right)}^{100-r}{\left(4\frac{1}{5}\right)}^r$

$={100}_{cr}{5}^{25-\frac{r}{4}}4\frac{r}{5}$

For rational term ${}^{\prime }{r}^{\prime }$ must be multiple of ${}^{\prime }{20}^{\prime }$

$\therefore r=0,20,40,60,80,100$

$\therefore$ Number of rational terms $=6$