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Q.
The number of irrational terms in the expansion of $\left(4^{1 / 5}+7^{1 / 10}\right)^{45}$ is
Binomial Theorem
Solution:
Total number of terms in the expansion of
$\left(4^{1 / 5}+7^{1 / 10}\right)^{45}$ is $45+1$, i.e., $46$ .
The general term in the expansion is
$T_{r+1}={ }^{45} C_{r} \cdot 4^{\frac{45-r}{5}} \cdot 7^{\frac{r}{10}}$
$T_{r+1}$ is rational if $r=0,10,20,30,40$.
$\therefore $ Number of rational terms $=5$.
$\therefore $ Number of irrational terms $=46-5=41$.