e^0.5 is larger.

Recall that p^0 = 1 for any positive value of p.
Now, there are useful results to know from here.

Suppose an expression is written as p^n.

If p > 1, then any positive power n will result in p^n being bigger than p itself. This means that e^0.5 > e^0, such that e^5 is bigger than 1.

[If the base p is greater than 1, then the positive powers serve to amplify the value of the base, think for example 2^5 = 2 x 2 x 2 x 2 x 2 and the value gets bigger and bigger]

if 0 < p < 1, then any positive power n will result in p^n being smaller than p itself. This means that 0.5^e < 0.5^0, such that 0.5^e < 1.

[Similarly, if the base p is between 0 and 1, then the positive powers serve to reduce the value of the base towards zero, think for example 0.5^5 = 0.5 x 0.5 x 0.5 x 0.5 x 0.5 and the value gets smaller and smaller]

So, it's pretty clear by now that 0.5^e < 1 < e^0.5, so e^0.5 is the larger number.