Answer
$at$ and $(2ax)^{\frac{1}{2}}$ have the dimension of speed.
Work Step by Step
Dimension of a (acceleration): $\frac{[L]}{[T]^{2}}$.
Dimension of t (time): [T].
Dimension of x (distance): [L].
Dimension of a speed: $\frac{[L]}{[T]}$.
The numbers 2 and $\frac{1}{2}$ have no dimensions.
(a) $\frac{1}{2}at^{2}=\frac{[L]}{[T]^{2}}[T]^{2}=[L]$.
$\frac{1}{2}at^{2}$ has not the dimension of a speed.
(b) $at=\frac{[L]}{[T]^{2}}[T]=\frac{[L]}{[T]}$.
$at$ has the dimension of a speed.
(c) $(\frac{2x}{a})^{\frac{1}{2}}=(\frac{[L]}{\frac{[L]}{[T]^{2}}})^{\frac{1}{2}}=([T]^{2})^{\frac{1}{2}}=[T]$.
$(\frac{2x}{a})^{\frac{1}{2}}$ has not the dimension of a speed.
(d) $(2ax)^{\frac{1}{2}}=(\frac{[L]}{[T]^{2}}[L])^{\frac{1}{2}}=(\frac{[L]^{2}}{[T]^{2}})^{\frac{1}{2}}=\frac{[L]}{[T]}$.
$(2ax)^{\frac{1}{2}}$ has the dimension of a speed.
