$T(n)=\sqrt2\ T\left ( \dfrac{n}{2} \right )+\sqrt{n}$
$\downarrow$
$\sqrt{2}\left (\sqrt{2}\ T\left ( \dfrac{n}{2^2} \right )+\sqrt{\dfrac{n}{2}} \right )+\sqrt{n}$
$(\sqrt{2})^2\ T\left ( \dfrac{n}{2^2} \right )+\sqrt{n}+\sqrt{n}$
.
.
.
$(\sqrt{2})^k\ T\left ( \dfrac{n}{2^k} \right )+k\sqrt{n}$
$\dfrac{n}{2^k}=1$
$k=log_{2}n$
$(\sqrt{2})^{log_{2}n}\ T\left ( \dfrac{n}{2^{log_{2}n}} \right )+log_{2}n\sqrt{n}$
$\sqrt{n}\ T\left ( 1 \right )+log_{2}n\sqrt{n}$
$\sqrt{n}(1+log_{2}n)$
$Ans:A$