Step 1: Treat the del operator as a vector operator:
The del operator is written as $\nabla = \left(\dfrac{\partial}{\partial x}, \dfrac{\partial}{\partial y}, \dfrac{\partial}{\partial z}\right)$, and divergence is formed by taking the dot product of $\nabla$ with a vector field, so $\nabla \cdot \vec{F} = \nabla_x F_x + \nabla_y F_y + \nabla_z F_z$ using this component notation.
Step 2: Use the distributive property of the dot product:
For any three vectors, the dot product distributes over vector addition, meaning $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$. Since divergence is defined through a dot product between $\nabla$ and the vector field, this same distributive property applies directly.
Step 3: Apply this property to the given expression:
Treating $\nabla$ as the vector $\vec{a}$, and $\vec{U}$ and $\vec{V}$ as $\vec{b}$ and $\vec{c}$, the distributive property gives $\nabla \cdot (\vec{U} + \vec{V}) = \nabla \cdot \vec{U} + \nabla \cdot \vec{V}$ directly, without needing to expand each component individually.
Step 4: Rule out the curl based options:
Options A and B both use the cross product symbol $\nabla \times$, which defines curl rather than divergence. Curl of a vector field returns another vector field, while the question asks specifically about divergence, which returns a scalar, so options A and B cannot be correct regardless of the sign used.
Step 5: Confirm the correct option:
Between the remaining choices, option D introduces an incorrect subtraction that does not arise from the dot product distributive law, while option C correctly reflects the addition, confirming that option C is the right choice.
Final Answer:
\[ \boxed{\nabla \cdot (\vec{U} + \vec{V}) = \nabla \cdot \vec{U} + \nabla \cdot \vec{V}} \]