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

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none of these

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Solution

The correct option is D 41
Total number of terms in the expansion of ${(4^{\frac{1}{5}} + 7^{\frac{1}{10}})}^{45}$ is $45 + 1 = 46$
The general term in the expansion is
$T_{r + 1} =^{45} C_{r} \times 4^{\frac{45 - r}{5}} \times 7^{\frac{r}{10}}$
$T_{r + 1}$ is rational if $r = 0, 10, 20, 30, 40$
$∴$ Number of rational terms $= 5$
$∴$ Number of irrational terms $= 46 - 5 = 41$

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