Let's simplify the expression step by step:

The expression is:

$\sqrt{49} + 4\sqrt{-9} + \sqrt[3]{512}$

Step 1: Simplifying $\sqrt{49}$

$\sqrt{49} = 7$

Step 2: Simplifying $4\sqrt{-9}$

Here, we encounter a negative square root, $\sqrt{-9}$. This involves imaginary numbers.

$\sqrt{-9} = 3i$ So, $4\sqrt{-9} = 4 \times 3i = 12i$

Step 3: Simplifying $\sqrt[3]{512}$

We take the cube root of 512: $\sqrt[3]{512} = 8$

Final Expression:

Combining all the simplified terms: $7 + 12i + 8 = 15 + 12i$

Thus, the simplified expression is:

$15 + 12i$

Do you want more details or have any questions?

Here are some related questions:

1. What are the general rules for simplifying square roots and cube roots?
2. How do imaginary numbers impact algebraic simplifications?
3. What is the difference between square roots and cube roots?
4. Can you explain why $\sqrt{-1} = i$ is used in mathematics?
5. What are other examples of expressions involving imaginary numbers?

Tip: Remember that when dealing with square roots of negative numbers, you always bring in the imaginary unit $i$.