Given relation

$z = a^2 x^3 y^{1/2}$

where $a$ is a constant.

Rule for percentage error

For a quantity of the form $Q = A^m B^n,$ the maximum percentage error is $\frac{\Delta Q}{Q} \times 100 = |m|\left(\frac{\Delta A}{A} \times 100\right) + |n|\left(\frac{\Delta B}{B} \times 100\right).$

Since $a$ is a constant, it contributes no error.

Apply to $z$

$z = a^2 x^3 y^{1/2}$

So, $\%\text{ error in } z = 3(\%\text{ error in }x) + \frac{1}{2}(\%\text{ error in }y).$

Substitute the given values

$\%\text{ error in }x = 4\%,$ $\%\text{ error in }y = 12\%.$

Therefore, $\%\text{ error in } z = 3(4) + \frac{1}{2}(12)$ $= 12 + 6$ $= 18\%.$

Final answer

$\boxed{18}$

Comparison with stored correct answer

Stored correct answer = $18$

My derived answer also = $18$

Hence, the answer agrees with the stored correct answer.