To find the related square root function \( p(x) \) for the quadratic equation \( h(x) = 4x^2 \) with the given domain of \( (0, \infty) \), we start by recognizing that the square root function is typically derived from solving for \( x \) in terms of \( h(x) \).
First, let's express \( h(x) \) using the square root:
\[ y = h(x) = 4x^2 \]
To solve for \( x \) in terms of \( y \):
\[ y = 4x^2 \]
Dividing both sides by 4 gives:
\[ \frac{y}{4} = x^2 \]
Taking the square root of both sides, we get:
\[ x = \sqrt{\frac{y}{4}} \]
This simplifies to:
\[ x = \frac{1}{2} \sqrt{y} \]
Now, to find the standard form of the square root function \( p(x) \), we need to express \( x \) in terms of \( y \). We can rearrange the equation as follows:
\[ p(y) = \frac{1}{2} \sqrt{y} \]
To rewrite this in the form \( p(x) = \sqrt{ax} \), we recognize that we can let \( x \) represent \( y \). Thus we can express \( p(x) \) as:
\[ p(x) = \sqrt{4} \cdot \frac{1}{2} \sqrt{x} = \sqrt{4} \cdot \sqrt{\frac{x}{4}} = \sqrt{4} \cdot \frac{1}{2} \cdot \sqrt{x} = \sqrt{x} \]
In the form \( p(x) = \sqrt{ax} \), we can conclude that \( a = \frac{1}{4} \).
Therefore, the answer to fill in the blank for \( a \) is:
\[ a = 4 \]
Thus, the function \( p(x) \) can be written as:
\[ p(x) = \sqrt{4x} \]
So, the final answer is:
\[ p(x) = \sqrt{4x} \quad \text{where } a = 4. \]