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The number of irrational terms in the expansion of  (\sqrt[8]{5}+\sqrt[6]{2})^{100} \text { is }

Option: 1

97


Option: 2

98


Option: 3

96


Option: 4

99

 


Answers (1)

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T_{r+1}={ }^{100} C_r 5^{\frac{100-r}{8}} \cdot 2^{\frac{r}{6}}

As 2 and 5 are co-primeT_{r+1} will be rational if 100-r  is multiple of 8 and r is multiple of 6 also 0 \leq r \leq 100 

\therefore r=0,6,12 \ldots \ldots .96 \ldots 100-r=4,10,16 \ldots . .100...(1)

But 100-r is to be multiple of 8

So,  100-r=0,8,16,24, \ldots .96...(2)

Common terms in (i) and (ii) are 16, 40, 64, 88.

r=84,60,36,12 give rational terms The number of irrational terms  = 101 – 4 = 97.

 

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