To determine which expression is in its simplest form, let's simplify each of the given expressions carefully:
Expression A:
[tex]\[ 4x \sqrt{2xy} \][/tex]
This expression is already simplified to its lowest terms because 4, [tex]\( x \)[/tex], and [tex]\( \sqrt{2xy} \)[/tex] are treated as separate factors.
Expression B:
[tex]\[ 3a^2 \sqrt{4b} \][/tex]
First, recognize that [tex]\(\sqrt{4b}\)[/tex] can be simplified:
[tex]\[ \sqrt{4b} = \sqrt{4} \cdot \sqrt{b} = 2\sqrt{b} \][/tex]
So the expression becomes:
[tex]\[ 3a^2 \cdot 2\sqrt{b} = 6a^2 \sqrt{b} \][/tex]
Expression C:
[tex]\[ c^3 d \sqrt{3d^3} \][/tex]
Here, we can simplify [tex]\(\sqrt{3d^3}\)[/tex]:
[tex]\[ \sqrt{3d^3} = \sqrt{3} \cdot \sqrt{d^3} = \sqrt{3} \cdot d^{3/2} \][/tex]
The expression becomes:
[tex]\[ c^3 d \cdot \sqrt{3} \cdot d^{3/2} = c^3 \cdot d^{1 + 3/2} \cdot \sqrt{3} = c^3 \cdot d^{5/2} \cdot \sqrt{3} = c^3 d^{5/2} \sqrt{3} \][/tex]
Expression D:
[tex]\[ 14s \sqrt{st^2} \][/tex]
Simplify [tex]\(\sqrt{st^2}\)[/tex]:
[tex]\[ \sqrt{st^2} = \sqrt{s} \cdot \sqrt{t^2} = \sqrt{s} \cdot t \][/tex]
So the expression becomes:
[tex]\[ 14s \cdot \sqrt{s} \cdot t = 14 \cdot s \cdot \sqrt{s} \cdot t = 14s^{3/2}t \][/tex]
Given the simplest forms of each expression, expression A (4x \sqrt{2xy}) is already in its simplest form and does not simplify any further.
Thus, the correct choice is:
[tex]\[ \boxed{4x \sqrt{2xy}} \][/tex]
