To evaluate \(\sqrt{-72}\), we need to work with complex numbers since the square root of a negative number is not defined within the set of real numbers.

### Step-by-Step Solution:

1. **Factor out the negative sign:**
   \[
   \sqrt{-72} = \sqrt{-1 \times 72}
   \]

2. **Separate the square roots:**
   \[
   \sqrt{-1 \times 72} = \sqrt{-1} \times \sqrt{72}
   \]

3. **Simplify \(\sqrt{-1}\) as \(i\), the imaginary unit:**
   \[
   \sqrt{-1} = i
   \]

   So, the expression becomes:
   \[
   \sqrt{-72} = i \times \sqrt{72}
   \]

4. **Simplify \(\sqrt{72}\):**
   - Factor 72 into its prime factors: \(72 = 2^3 \times 3^2\).
   - Use the property of square roots: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).

   Thus:
   \[
   \sqrt{72} = \sqrt{2^3 \times 3^2} = \sqrt{2^2 \times 2 \times 3^2} = \sqrt{(2 \times 3)^2 \times 2} = \sqrt{36 \times 2}
   \]

   Then:
   \[
   \sqrt{72} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}
   \]

5. **Combine the results:**
   \[
   \sqrt{-72} = i \times 6\sqrt{2} = 6i\sqrt{2}
   \]

### Final Answer:

\[
\sqrt{-72} = 6i\sqrt{2}
\]

Would you like more details or have any questions about this solution?

Here are 5 related questions to explore further:

1. What is the square root of a negative number, such as \(\sqrt{-9}\)?
2. How do you simplify square roots of non-perfect squares like \(\sqrt{50}\)?
3. What are imaginary numbers, and why are they important in mathematics?
4. Can you explain how to multiply and divide complex numbers?
5. How do you represent complex numbers in polar form?

**Tip:** When dealing with square roots of negative numbers, always factor out \(-1\) first and use the imaginary unit \(i\).

Learn how to evaluate sqrt(-72) using complex numbers and the imaginary unit (i). This problem explores the concept of square roots of negative numbers, providing a detailed step-by-step solution. Suitable for students in Grades 10-12 studying advanced algebra and complex numbers.

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