The number of irrational terms in the expansion of  ${(4^{1/5}+7^{1/10})}^{45}$ is

Total number of terms in the expansion of ${\left(4^{\frac{1}{5}}+7^{\frac{1}{10}}\right)}^{45} \text{is }45+1$ i.e., 46

The general term in the expansion is ${\mathrm{T}}_{\mathrm{r}+1}{=}^{45}{\mathrm{C}}_{\mathrm{r}}\cdot 4^{\frac{45-\mathrm{r}}{5}}\cdot 7^{\frac{\mathrm{r}}{10}}$

${\mathrm{T}}_{\mathrm{r}+1}$ is rational if r = 0, 10, 20, 30, 40

 Number of rational terms = 5.
∴ Number of irrational terms = 46 – 5 = 41.