The number of integral terms in the expansion of $$\left(3^{\frac{1}{2}} + 5^{\frac{1}{4}}\right)^{680}$$ is equal to _______

The general term in the expansion of $$\left(3^{1/2} + 5^{1/4}\right)^{680}$$ is:

$$T_{r+1} = \binom{680}{r} \cdot 3^{(680-r)/2} \cdot 5^{r/4}$$

For the term to be an integer, both exponents must be non-negative integers:

Condition 1: $$\frac{680-r}{2}$$ is a non-negative integer $$\Rightarrow r$$ is even

Condition 2: $$\frac{r}{4}$$ is a non-negative integer $$\Rightarrow r$$ is divisible by 4

Both conditions are satisfied when $$r$$ is a multiple of 4.

$$r = 0, 4, 8, 12, \ldots, 680$$

Number of values = $$\frac{680}{4} + 1 = 170 + 1 = 171$$