[ Pascal's Triangle and Newton's Binomial ]

How many rational terms are contained in the expansion of

a) $(\sqrt{2}+\sqrt[4]{3})^{100}$

b) $(\sqrt{2}+\sqrt[8]{3})^{300}$?
a) For the number $(\sqrt{2})^{100-k}(\sqrt[4]{3})^{k}$ to be rational, $k$ must be a multiple of 4. There are 26 such integers: $0, 4, 8$, ..., 96, 100.

b) For the number $(\sqrt{2})^{300-k}(\sqrt[3]{3})^{k}$ to be rational, $k$ must be a multiple of 3, and $300-k$ must be even. Therefore, $k$ must be a multiple of 6. Among the integers from 0 to 300, there are $300: 6+1=51$ such numbers.

## Answer

a) 26; b) 51.