First slide

Binomial theorem for positive integral Index

Question

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

Moderate

A

40
B

5
C

41
D

none of these

Solution

Total number of terms in the expansion of ${(4^{\frac{1}{5}} + 7^{\frac{1}{10}})}^{45} is 45 + 1$ 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
Number of rational terms = 5.
∴ Number of irrational terms = 46 – 5 = 41.

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