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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.

Write the expression in terms of i and simplify.

Explore how to simplify expressions using imaginary numbers with a detailed step-by-step guide. This example simplifies the product of square roots of negative numbers, highlighting the use of the imaginary unit i and its properties. Suitable for high school students learning about complex numbers.

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