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  4. The number of irrational terms in the expansion of (3(1/8)+5(1/4))84 is

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Q. The number of irrational terms in the expansion of $ \left(3^{\frac{1}{8}}+5^{\frac{1}{4}}\right)^{84}$ is

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Solution:

$T_{r+1} = ^{84}C_{r} \left(3^{\frac{1}{8}}\right)^{84-r} \left(5^{\frac{1}{4}}\right)^{r} = ^{84}C_{r} \cdot3^{\left(\frac{84-r}{8}\right)}. 5^{\frac{r}{4}}$
$T_{r+1}$ is rational for r = 4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, Number of Irrational terms = 85 - 11 = 74