Recall the definition of a square root in terms of exponents.

The square root of a number $a$ can be written as $a$ raised to the power of $\frac{1}{2}$.

The options are:
[A] $a^{\frac{1}{2}}$
[B] $a^{-2}$
[C] $a^{2}$
[D] $a^{-\frac{1}{2}}$

The expression $a^{\frac{1}{2}}$ matches option [A].

The final answer is $\text{A}$.