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  4. The sum of rational terms in (√2+√[3]3+√[6]5)10 is

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Q. The sum of rational terms in $(\sqrt{2}+\sqrt[3]{3}+\sqrt[6]{5})^{10}$ is

Binomial Theorem

Solution:

General term $ = \frac{10!}{\alpha ! \beta ! \gamma !} 2^{\alpha/2} 3^{\alpha/3} 5^{\gamma/6}$
For rational terms
$\alpha=0,2,4,6,8,10$
$\beta=0,3,6$
$\gamma=0,6$
Hence, possible sets $=(4,6,0),(4,0,6),(10,0,0)$
Hence, there are $3$ rational terms.
$\therefore$ Required sum $=\frac{10 !}{4 ! 6 !} 2^{2} 3^{2}+\frac{10 !}{4 ! 6 !} 2^{2} 5+\frac{10 !}{10 !} 2^{5}$
$=12632$