The given expression is:

$\frac{\sqrt{-16}}{\sqrt{-48}}$

Since square roots of negative numbers are involved, we can express them using imaginary numbers. Recall that $\sqrt{-1} = i$, where $i$ is the imaginary unit.

Step 1: Simplify the square roots

$\sqrt{-16} = \sqrt{16} \times \sqrt{-1} = 4i$ $\sqrt{-48} = \sqrt{48} \times \sqrt{-1} = \sqrt{16 \times 3} \times i = 4\sqrt{3}i$

Step 2: Simplify the fraction

Now the expression becomes: $\frac{4i}{4\sqrt{3}i}$

Step 3: Cancel the common terms

The $4i$ cancels out, leaving: $\frac{1}{\sqrt{3}}$

Step 4: Rationalize the denominator

To eliminate the square root in the denominator, multiply both the numerator and denominator by $\sqrt{3}$: $\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

Thus, the simplified result is: $\frac{\sqrt{3}}{3}$

Would you like more details or have any questions?

Here are 5 related questions you might explore:

1. What is the meaning of imaginary numbers?
2. How do you simplify expressions involving square roots?
3. How do you rationalize a denominator?
4. What are complex numbers and how are they used in math?
5. How do you perform arithmetic with complex numbers?

Tip: Always check for opportunities to simplify or cancel terms when dealing with fractions.