The given expression is:

$\frac{-3 + \sqrt{-27}}{3}$

To simplify this, follow these steps:

1. Simplify the square root of $-27$: $\sqrt{-27} = \sqrt{-1 \times 27} = \sqrt{-1} \times \sqrt{27} = i \times \sqrt{27}$ We know that $i$ represents the imaginary unit, and $\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$.

   So: $\sqrt{-27} = 3i\sqrt{3}$

2. Substitute back into the expression: $\frac{-3 + 3i\sqrt{3}}{3}$

3. Simplify the fraction: We can now divide both terms in the numerator by 3: $\frac{-3}{3} + \frac{3i\sqrt{3}}{3} = -1 + i\sqrt{3}$

Thus, the simplified expression is: $-1 + i\sqrt{3}$

Would you like more details on any of these steps or have additional questions?

Here are 5 related questions to consider:

1. What is the significance of the imaginary unit $i$ in complex numbers?
2. How do you simplify square roots involving negative numbers?
3. What are the steps to perform division of complex numbers?
4. Can you perform similar operations with other complex numbers?
5. How would you graph the number $-1 + i\sqrt{3}$ on the complex plane?

Tip: When simplifying square roots of negative numbers, always factor out $\sqrt{-1} = i$ first before proceeding with the rest of the square root.