According to Extended Masters Theorem,
$T(n) = aT(n/b)+n^k\log^p n, a\geq 1, b>1, k \geq 0$ and $p$ can be a real number
- if $a> b^k$
$T(n)=\Theta \left (n^{\log_b a} \right)$
- if $a = b^k$
- if $p>-1, T(n) = \Theta \left(n^{\log_b a} \log^{p+1} n\right)$
- if $p=-1, T(n) = \Theta \left(n^{ \log_b a} \log \log n\right)$
- if $p<-1, T(n) = \Theta\left(n^{\log_b a} \right)$
- if $a< b^k$
- if $p \geq 0, T(n) = \Theta \left(n^k \log^pn \right)$
- if $p<0, T(n) = O \left(n ^k\right)$
in the given question, $a=1,b=2,k=1/2$ and $p=0$
here $a<b^k$ and $p=0$ (case 3(a))
so $T(n) = \Theta\left(n^k \log^pn \right)$
$T(n) = \Theta \left(n ^{1/2}\right)$
Alternatively we can apply Master theorem as given in Cormen,
$$T(n) = aT(n/b)+ f(n)$$
Here, $a = 1, b = 2, f(n) = n^{1/2}, f(n) = \Omega \left( n^{\log_b a + \epsilon} \right) \implies n^{1/2} = \Omega\left( n^{\log_2 1 + \epsilon} \right)$, is true for any $\epsilon < \frac{1}{2}$, and thus we have some positive $\epsilon$. So, Case 3 of Master theorem. But Case 3 also requires regularity condition which states, $$af\left(\frac{n}{b} \right)\leq c f(n)$$ for some $c < 1$.
Here, we get
$af\left(\frac{n}{b} \right) = f\left(\frac{n}{2} \right) = \frac{n^{1/2} }{ \sqrt 2} \leq c f(n)$,
for any $c \leq \frac{1}{\sqrt 2}.$
So, regularity condition also satisfied and we get $T(n) = \Theta (f(n)) = \Theta \left(n^{1/2}\right)$.