Let's simplify the two expressions in the image.

Expression 1:

$\frac{\sqrt{-81}}{\sqrt{-3}}$

• The square root of a negative number introduces the imaginary unit $i$, where $i = \sqrt{-1}$.
• Simplify the numerator: $\sqrt{-81} = \sqrt{81} \cdot \sqrt{-1} = 9i$.
• Simplify the denominator: $\sqrt{-3} = \sqrt{3} \cdot \sqrt{-1} = \sqrt{3}i$.
• Now, divide the two: $\frac{9i}{\sqrt{3}i} = \frac{9}{\sqrt{3}} = \frac{9\sqrt{3}}{3} = 3\sqrt{3}.$

Expression 2:

$\sqrt{-2} \cdot \sqrt{32}$

• Again, introduce $i$ for the negative square root. $\sqrt{-2} = \sqrt{2} \cdot i$.
• Simplify the second square root: $\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}$.
• Now, multiply the two: $(\sqrt{2} \cdot i) \cdot (4\sqrt{2}) = 4 \cdot i \cdot (\sqrt{2} \cdot \sqrt{2}) = 4i \cdot 2 = 8i.$

Final Answers:

1. $\frac{\sqrt{-81}}{\sqrt{-3}} = 3\sqrt{3}$
2. $\sqrt{-2} \cdot \sqrt{32} = 8i$

Would you like more details on the steps or have any further questions?

Here are 5 related questions to expand on this:

1. How do you simplify expressions involving imaginary numbers and square roots?
2. What is the significance of the imaginary unit $i$ in mathematics?
3. How can you rationalize denominators when radicals are involved?
4. Can complex numbers be used to solve real-world problems?
5. What are the properties of square roots of negative numbers?

Tip: Always express square roots of negative numbers in terms of the imaginary unit $i$ to simplify complex expressions.