What does it mean "beat"? From the context, I'm deriving that in computational complexity it is said that a polynomial time is always better than an exponential time.
Now, if you look here a polynomial time is defined as $O(n^c)$, while an exponential time is defined as $O(c^n)$, where $c$ is a positive constant, and $n$ increases without bound.
Your example does not involve really an exponential and a polynomial. It is a polynomial and "something else". Since the $\log$ balances the exponential. Therefore it is not an exponential.
In order to see which function grows larger, you can even draw them. Just for the sake of the argument let's consider the inputs: 1, 10, 100 for $n$.
Case $n = 1$
Then you have that $n^\frac{1}{2}$ is $1$, while $2^\sqrt{\log n}$ is $1$.
Case $n = 10$
Then you have that $n^\frac{1}{2}$ is $3.16$, while $2^\sqrt{\log n}$ is $3.53$.
Case $n = 100$
Then you have that $n^\frac{1}{2}$ is $7.8$, while $2^\sqrt{\log n}$ is ~$6$.
So $n^\frac{1}{2}$ grows significantly faster. That is mainly due to the logarithm over the $2$.
So to reply to the question: which one of the two function is greater? It is up to which $n$. The first function grows faster, but for small values of $n$, the second function is bigger. The more $n$ grows, the more the first function will grow, therefore being greater than the second one.