The problem involves simplifying the expression $\sqrt{-16} \cdot \sqrt{-36}$ in terms of $i$, the imaginary unit.

Steps:

1. Recall that $\sqrt{-a} = i\sqrt{a}$ for any positive real number $a$. Using this property: $\sqrt{-16} = i\sqrt{16} = i \cdot 4 = 4i$ $\sqrt{-36} = i\sqrt{36} = i \cdot 6 = 6i$

2. Multiply the two results: $\sqrt{-16} \cdot \sqrt{-36} = (4i) \cdot (6i) = 24i^2$

3. Simplify using $i^2 = -1$: $24i^2 = 24(-1) = -24$

Thus, the simplified expression is $-24$.

Would you like any further explanation, or do you have any questions?

Here are five related questions to expand your understanding:

1. What is the general formula for simplifying square roots of negative numbers?
2. How do you simplify expressions with powers of $i$ (such as $i^3$, $i^4$)?
3. What is the significance of $i^2 = -1$ in complex numbers?
4. Can the product of two imaginary numbers always result in a real number?
5. How would you add or subtract two complex numbers?

Tip:

Always remember the pattern of powers of $i$: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$, and then the cycle repeats.