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Let's solve the problem step by step.

The expression given is \( \sqrt{-80} \).

1. First, break down the square root of a negative number:
   \[
   \sqrt{-80} = \sqrt{-1 \times 80} = \sqrt{-1} \times \sqrt{80}
   \]
   Since \( \sqrt{-1} = i \) (where \( i \) is the imaginary unit), the expression becomes:
   \[
   i \times \sqrt{80}
   \]

2. Now simplify \( \sqrt{80} \):
   \[
   \sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}
   \]

3. Combine the results:
   \[
   i \times 4\sqrt{5} = 4i\sqrt{5}
   \]

Thus, \( \sqrt{-80} = 4i\sqrt{5} \).

The correct answer is **\( 4i\sqrt{5} \)**, which corresponds to the first option.

Would you like any further details or have questions about this explanation?

Here are 5 related questions for deeper understanding:
1. How do you simplify square roots of negative numbers in general?
2. What is the imaginary unit \( i \) and how does it relate to negative square roots?
3. How can you decompose other square roots like \( \sqrt{-50} \)?
4. What are the rules for simplifying radical expressions involving imaginary numbers?
5. How can you apply this technique to solving quadratic equations with complex roots?

**Tip**: Always break down the number inside the square root into its prime factors to simplify it easily.

Which expression is equivalent to sqrt(-80)?

This problem demonstrates how to simplify the square root of a negative number, specifically sqrt(-80), into an expression involving an imaginary unit. It covers key concepts like imaginary numbers and simplifying radical expressions, suitable for Grades 9-11.

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