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To simplify the expression \( \sqrt{-4} + 3\sqrt{-16} \), we need to handle the square roots of negative numbers by expressing them in terms of imaginary numbers.

1. **Rewrite the square roots of negative numbers using \( i \):**
   - \( \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i \).
   - \( \sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i \).

2. **Substitute these values back into the expression:**
   \[
   \sqrt{-4} + 3\sqrt{-16} = 2i + 3 \cdot 4i.
   \]

3. **Simplify further by performing the multiplication:**
   \[
   2i + 3 \cdot 4i = 2i + 12i.
   \]

4. **Combine like terms:**
   \[
   2i + 12i = 14i.
   \]

So, the simplified expression is:
\[
\sqrt{-4} + 3\sqrt{-16} = 14i.
\]

Would you like further explanation on any of these steps?

### Related Questions:
1. How do we simplify expressions involving the square roots of negative numbers?
2. What are imaginary and complex numbers, and how are they used?
3. How would this expression change if the square roots were positive?
4. Can you explain why we use \( i \) when dealing with square roots of negative numbers?
5. What other operations can we perform with imaginary numbers?

**Tip:** Remember, the imaginary unit \( i \) is defined as \( i = \sqrt{-1} \), which makes it possible to simplify square roots of negative numbers.

Learn how to simplify expressions involving square roots of negative numbers, such as \( \sqrt{-4} + 3\sqrt{-16} \), using imaginary numbers. This problem demonstrates how to rewrite and combine terms with imaginary units, making it ideal for high school math students.

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